These are Zixuan’s notes for Part IA – Differential Equations at the University of Cambridge in 2025. The notes are not endorsed by the lecturers or the University, and all errors are my own.

The latest version of this document is available at academic.micfong.space. Please direct any comments to my CRSid email or use the contact details listed on the site.

This document is typeset using Typst. Almost all figures are created using Inkscape and Mathematica.

Differential equations appear in almost all branches of science and applied mathematics.

For example, we have the following differential equation,

which relates the rate of change of position , the dependent variable, with time , the independent variable.

The main purpose of this course is to solve such equations.

1 Differentiation

1.1 Introduction

Informally, if , then can be made arbitrarily close to by making sufficiently close to . Nonetheless, we do not require . [See IA Analysis I for a more formal discussion.]

We can higher derivatives for sufficiently smooth functions. For example,

To refer to the th derivative, we write

1.2 Big O and Little O

If we wish to compare the behavior of functions close to a limiting point , we can use order parameters. There are two kinds of them: Big O and Little O.

1. Big O — “can be bounded by”

   Definition 1.2 (Big O)

   • if is finite, then is as if and such that with , we have

     We often write It follows that is bounded as .

     e.g. if , then and .

   

   e.g. as since .

   • if , then is as if and such that ,

     e.g. as since ,

1. Little O — “much smaller than”

   Definition 1.3 (Little O)

   is as if such that ,

   If in vicinity of (regardless of the behavior at ), equivalently

   We often write .

   e.g. as , since .

   e.g. as .

Order parameters are useful to classify the remainder terms before taking limits. So we can write

Therefore as .

Hence

as . This result extends to Theorem 1.10 (Taylor’s Theorem).

1.3 Rules for Differentiation

Consider By repeated applying the product rule, we have

c.f. Pascal’s triangle, we can generalize this into Leibniz’s Rule.

1.4 Taylor Series

Note that are the partial sums of .

With , Theorem 1.10 (Taylor’s theorem) gives

This is to say, that provides a local approximation to in the vicinity of with error or .

Therefore, for a given target accurarcy at , this can be used to specify how large must be.

1.5 L’Hôpital’s Rule

L’Hôpital’s rule allows us to deal with limits of indeterminate forms.

2 Integration

We have seen the definition of integration in terms of area under a curve and inverse of differentiation. We shall review what these really mean.

2.1 Integrals as Riemann Sums

This section talks about the area under a curve idea.

How are we sure this limit evaluates to the area under a curve idea? We shall consider one rectangle at first.

We will need to borrow this following idea from IA Analysis I for now:

Hence, if is differentiable, then by Taylor’s theorem

Hence,

Therefore, the total area under curve from to is

2.2 Fundamental Theorem of Calculus (FTC)

This section talks about the inverse of differentiation idea.

2.3 Integration Techniques

2.3.1 Integration by Substitution

If the integrand contains a function of a function, it might help to substitute for the inner function.

Another important class of substitutions are trigonometric substitutions. These make use of the following identities:

We can use the following table as a substitution reference.

Integrand contains	Substitution

2.3.2 Integration by Parts

Recall the product rule

we can derive the integration by parts technique

This integration by parts method also works for inverse trignometric functions and inverse hyperbolic functions.

3 Partial Differentiation

3.1 Functions of Several Variables

We shall now generalize to functions of more than one independent variables, which are also called multivariate functions.

A way of representing these functions is using a contour plot.

The slope of a point on the plot depends on the direction. Firstly, let us consider what happens along the coodinate directions.

3.2 Partial Derivatives

Partial derivatives ar ederivatives of multivariate functions with respect to one variable, while holding the other fixed.

Note that in Example 3.3, it seems that

This is indeed the case under certain conditions.

If is a multivariate function with continuous 2nd derivatives, then

if the conditions of Schwarz’s theorem is satisfied.

3.3 Multivariate Chain Rule

Give path and a function , consider along the path.

Thus, for the path , we have

If instead we parameterize path by coordinate , so we are left with , then

We can also reach the integral form of chain rule:

For ,

so the final result does not depend on a particular path for given endpoints.

3.4 Applications of Multivariate Chain Rule

3.4.1 Change of Variables

It is often useful to write a differential equation in a different coordinate system.

For example, we can change Cartesian coordinates to polar coordinates with

Think of as . From the chain rule,

3.4.2 Implicit Differentiation

For , we have

Consider , which represents a surface in 3D space. In this case, it implicitly defines

but we may not be able to find the solutions explicitly.

However, we can still evaluate derivatives like .

The reciprocal rule applies if the same variables are held fixed:

Therefore,

3.5 Differentiation of an Integral w.r.t. a Parameter

4 First-Order Linear Differential Equations

4.1 The Exponential Function

In order to introduce first-order linear ordinary differential equations, we first need to explore the exponential function.

Differentiaing the exponential function, we have

This allows us to define the exponential function in another way. We can define to be the solution of

with the initial condition .

Following this property, it suggests that we should write as , where

The inverse function is defined as the function where

It follows that

Hence

Hence, the eigenfunctions of the differential operator are of the form

4.2 First Order Linear Ordinary Differential Equations

4.2.1 Homogeneous Linear ODEs with Constant Coefficients

To specify unique solutions, it requires us to apply suitable initial conditions. [ conditions are needed for a th order ODE.]

For example, in Example 4.12, if we have , then .

4.2.2 Discrete Equations

It is sometimes useful to consider functions evaluated at discrete points.

Consider again with .

We can approximate this equation by discrete form at with and . With , we have

[This is called the Forward Euler scheme, which is not a great approximation numerically.]

Substituting this approximation into the original equation gives

Hence we have

Now take [this represents steps from to ] as . Then

which, thankfully, agrees with the continuous case.

4.2.3 Series Solutions

This is a powerful way to solve ODEs. Essentially, we are looking for soluions in the form of a power series,

where we will determine by substituting into the ODE.

4.3 Forced (non-homogeneous) ODEs

In a forced ODE, there are terms not involving the dependent variable or its derivatives. In this case, in general, is no longer a solution. To solve these equations, we should do the followings.

1. Find any solution of the forced equation, called a particular integral (PI) .

2. Write the general solution as , and find the complementary function (CF) that satisfies the corresponding homogeneous equation.

3. Combine and to get the general solution.

This method is general for linear ODEs.

4.3.1 Constant Forcing

4.3.2 Eigenfunction Forcing

The forcing term may also be an eigenfunction of the differential operator.

4.4 Non-constant Coefficients

The general form of such equations is

We can get the standard form by dividing both sides by (assuming ):

To solve these equations, we use an integrating factor (IF) . Multiplying our standard form by gives

We want the left-hand side to be , so we require by the product rule. This is a separable equation, so

Therefore,

which is unique up to an irrelevant constant factor. Hence, the original equation becomes

Let us get back to Example 4.15.

5 Nonlinear First-Order ODEs

Recall that a non-linear ODE is one in which the dependent variable [usually ] and its derivatives appear with exponents other than 1 or are multiplied together.

The general form of a first-order nonlinear ODE is

[The term in could be nonlinear, but it is not considered here.]

We have two special cases of nonlinear first-order ODEs that we can solve: separable equations and exact equations.

5.1 Separable Equations

These equations can be solved by integrating both sides.

5.2 Exact equations

If an ODE is exact, then using the multivariate chain rule, we have

So, we can solve exact equations by finding a function such that and .

Since partial derivatives commute, we have

This is a necessary condition but not sufficient for exactness:

5.3 Solution Curves and Isoclines

The general idea is that nonlinear ODEs are often impossible to solve in simple closed forms. We can still analyse the behaviour of solutions using graphical methods.

5.3.1 Solution Curves

Consider

Then, each initial condition (e.g. ) generates a distinct solution curve (trajectory).

We can still sketch these solution curves without actually solving the ODE.

The equation gives us the gradient of the solution through .

Note that the solution curves can’t cross if is single valued.

5.4 Fixed (Equilibrium) Points and Stability

These points often reveal important features of ODEs.

5.4.1 Perturbation Analysis and Stability

Let be a fixed point of [i.e. for all ]. Consider a small perturbation about the fixed point:

We want to analyze how behaves as .

Then

Linearize for small epsilon:

Not that this is a linear ODE in .

Note that if , then we need higher order terms to determine stability.

5.4.2 Autonomous Systems and Phase Portraits

First of all, these autonomous systems are separable. We can write

Hence, if is a solution, then so is for any constant .

Although this equation is separable, we may not be able to solve it in closed form. Consider, near a fixed point , we have

Hence, if

• , then the fixed point is stable.
• , then the fixed point is unstable.

5.5 Fixed Points in Discrete Equations

Note that the RHS is independent of , which is like an autonomous system in continuous ODEs.

We can also analyse the stability of fixed points in discrete equations using perturbation analysis. Let be a fixed point, and write , where .

Then,

Therefore, is

• stable if ,

• unstable if .

6 Higher Order Linear ODEs

We shall focus on 2nd order linear ODEs, but many methods are also applicable to higher order linear ODEs.

Different to the 1st order case, closed form solutions to 2nd order linear ODEs don’t always exist.

6.1 2nd Order ODEs with Constant Coefficients

The general form of a 2nd order linear ODE with constant coefficients is

where are constants and is a given function, and is the differential operator

which is linear.

We can exploit the linearity of to solve 2nd order linear ODEs.

1. Find the complementary functions that satisfy the homogeneous equation:

2. Find a particular integral that satisfies the non-homogenous equation:

3. A solution of the full equation is then given by

   since by linearity, .

A 2nd order ODE has two linearly independent complementary functions, so the general solution to the full equation is

One can compare the two definitions above with the definition of vectors.

6.2 Complementary Functions

Recall that

so is also an eigenfunction of , where

The complementary functions satisfy , i.e. are eigenfunctions of with eigenvalue .

Therefore, with satisfying the characteristic equation of :

There are two roots, , to the characteristic equation, which can be real or complex.

• Case 1:

  We have two linearly independent complementary functions:

  Hence, the most general complementary function is

  In the language of linear algebra, the space of complementary functions is a 2-dimensional vector space spanned by the basis .

  Remark. The roots may be complex, which will lead to oscillations.

• Case 2: (degenerate case)

  We only have one linearly independent complementary function . See the following examples to find the second complementary function.

General rule. If is a degenerate complementary function of linear ODE with constant coefficients, then

is a second linearly independent complementary function.

6.3 Homogeneous 2nd Order ODEs with Non-Constant Coefficients

The general form of this kind of ODE is

6.3.1 Second Complementary Function – Reduction of Order

We wish to find a 2nd solution given one solution .

We will try from the form

where is an unknown function to be determined. We have

Substituting into the ODE, we get

Therefore,

Let , then

This is a separable 1st order ODE for :

Integrating both sides, we get

Hence, we can integrate to find , and then find .

We shall see an example with constant coefficients first.

6.3.2 Phase Space

For an th order linear ODE

is determined by , and higher derivatives are determined by differentiating the ODE.

Hence, we can construct Taylor series about if are specified.

We say that the state of the system at any is fully specified by an -dimensional solution vector.

6.3.3 Wronskian and Linear Dependence

Recall that are linearly dependent if there exist constants , not all zero, such that

Hence, we can differentiate this equation times to get

so being linearly dependent implies that is linearly dependent.

From the above, if are linearly dependent, then for all .

It follows that if for some , then are linearly independent.

6.3.4 Abel’s Theorem

The geometric interpretation is that the solution vectors are either always collinear or never collinear in the phase space.

We can find without knowing the solutions explicitly.

Abel’s theorem can be generalised.

6.4 Linear Equidimensional ODEs

These ODEs are related to ODEs with constant coefficients.

6.4.1 Solving by Eigenfunctions

so is an eigenfunction of the operator with eigenvalue .

It suggests that we should look for complementary functions of the form . Substituting into the homogeneous equation, we get

Let the solutions be . Then the complementary functions are

if .

6.4.2 Solving by Substitution

Substitute , then

Substituting into the ODE, we get

Now we have a linear ODE with constant coefficients in the variable , which we can solve using the methods discussed earlier. The characteristic equation is

This is the same characteristic equation as before when we tried the eigenfunction method. The complementary functions are therefore, if ,

We can now deal with the degenerate case similarly as before.

6.5 Inhomogeneous (Forced) 2nd Order ODEs

We will discuss the methods to find particular integrals.

6.5.1 Constant Coefficient ODEs

We have the form

Now, use the following ansatz for depending on the form of :

	Try of the form
or
Polynomial of degree

We can determine the constants by substituting into the ODE. Since the ODE is linear, we can superpose terms.

If the forcing term involves a term that is in a complementary function, a resonance case occurs. In this case, we carry out detuning.

The general rule is that if the forcing term is a linear combination of linearly independent complementary functions, the particular integral is of the form

6.5.2 Equidimensional ODEs

Consider the equidimensional ODE

We have seen that the complementary functions are of the form

assuming .

If , we try a particular integral of the form

for and .

For the resonance cases or , then the particular integral is of the form

which follows from Section 6.4.2.

6.6 Variation of Parameters

This is a systematic method to find particular integrals given two linearly independent complementary functions.

Consider

with linearly independent complementary functions and .

We will use solution vectors and as a basis in phase space at any to write the solution vector for the particular integral.

We have

The components are

If we differentiate again, we get

Subsituting into the ODE, we have

Since are complementary functions, we have

Therefore,

Note that the second component of must be consistent with the derivative of the first component. Therefore, we have the additional constraint

We now have two equations for and :

We can solve for and :

We can therefore invert the fundamental matrix to get

Hence, we can integrate to find and :

The particular integral is therefore

Note that changing the lower limits of the integrals only adds multiples of complementary functions to .

6.7 Forced ODEs, Transients and Damping

In the diagram above, by Newton’s 2nd Law we have

Rearranging, we get

where are positive constants.

For and , we have simple harmonic motion at angular frequency

For convenience, we will add in a dimensionless time coordinate . Then we have . Therefore,

where

6.7.1 Free (Unforced) Response

The behavior is described by one dimensionless parameter . We have

The characteristic equation is

The roots are

Depending on the value of , we have three cases:

1. Light damping (underdamping) when .

   In this case, the roots are complex, so we can write

   So the general solution is

   where are constants.

   

   This is a oscillation at , which tends to as , with an exponentially decaying amplitude.

   The period is

2. Critical damping when .

   We have a degenerate root .

   The general solution is

   where are constants.

   

3. Heavy damping when .

   The roots are real and negative. WLOG take . Then

   The general solution is

   where are constants. [ indicates that the exponentials decay to zero as .] Note that , so the first term dominates the long-term behavior if present.

   

Unforced response decays eventually in all cases.

6.7.2 Forced Response

Initially, the behavior is determined by , which is called the transient response.

Over time, decays, and the behavior is dominated by the , called the steady-state response.

6.8 Impulses and Point Forces

Conside a system that experiences a sudden force between time and .

e.g. striking a mass on a spring, or a car going over a curb.

The equation can be in the form

which is a forced, damped oscillator.

It is mathamatically convienient to consider the limit of a sudden impluse, as .

We can integrate the ODE from to , and take the limit .

So we see that

and the velocity is discontinuous.

As only impluse matters for subsequent motion.

6.8.1 Dirac Delta Function

We shall formalise the idea of an impulsive force.

Consider a family of functions such that

and

So the impulsive force we considered earlier is:

One example of such a family is

[See Example Sheet 1, Q14 for normalisation.]

Note that the family of such functions is not unique, but for any family, yields the Dirac delta function.

6.8.2 Delta Function Forcing

Consider the ODE

assuming that and are continuous.

For and , we have the homogeneous equation

But there is a discontinuity in at :

Therefore,

which is referred to as a jump condition.

The general rule for higher order ODEs is that the highest-order term in ODE addresses the delta-function forcing.

6.8.3 Heaviside Step Function

By the fundamental theorem of calculus, we have

Note that functions get smoother as we integrate them.

6.8.4 Forcing with

Consider the ODE

assuming that and are continuous at .

So, we have

Therefore, and is continuous at . We have around . So, evaluating both sides around ,

The jump conditions for Heaviside function are

A typical situation is that for , and we are asked to find for . This leads to two arbitrary constants, and we can determine them using the two jump conditions.

Or alternatively, we can solve the ODE for and by matching at using the jump conditions.

7 Higher Order Discrete Equations

Consider a linear, discrete, 2nd order equation with constant coefficients of the form

where , , are constants.

We can solve this using similar methods to solving 2nd order ODEs, with general solution

where is the complementary function, and is a particular integral.

7.1 Complementary Functions

The complementary function must satisfy

We can try , since is an eigenfunction. This gives the characteristic equation

The general complementary function is

where , are the roots of the characteristic equation, and , are arbitrary constants.

7.2 Particular integrals

We can find particular integrals based on the form of :

8 Series Solutions

When we cannot find simple closed form solutions to ODEs, series solutions may be useful.

Consider

The feasibility of finding series solution around depends on nature of , , and around .

8.1 Classification of Singular Points

Compare this to an equidimensional equation

where is a regular singular point. We can state that regular singular points are no more singular than in equidimensional equations.

8.2 Method of Frobenius

The series solution method may fail completely at irregular singular points.

8.3 Second Solutions

Note that we are guaranteed to get one series solution about a regular singular point, but whether we get a second linearly independent such solution depends on the roots of the indicial equation, and .

1. If is not an integer, then we get two linearly independent series solutions, in the form

2. If is a non-zero integer, then we get one series solution involving the larger root, say .

   The second solution has the form:

   where may or may not be zero. is a constant determined in terms of and , so that we have two arbitrary constants as required.

3. If , our solutions is similar to case (2), but the logarithmic term is always present.

9 Multivariate Functions: Applications

In this section we will discuss

• directional derivatives
• extrema
• coupled systems of 1st order ODEs
• partial differential equations

9.1 Directional derivatives

Consider and a vector displacement .

The infinestimal change in along is given by

where we have defined the gradient operator

If we write where is a unit vector in the direction of , then we have

9.2 Stationary Points

There is always at least one direction where is zero at a given point, namely the direction parallel to the contours of at that point.

So stationary points are the points where

9.2.1 Types of Stationary Points

Note that contours cross at (and only at) saddle points.

9.3 Classification of Stationary Points

We shall consider how change in vicinity of a stationary point.

9.3.1 Taylor Series for Multivariate Functions

Consider how varies along the line

Along the line, is a function of , and we can use the usual Taylor series for single variable functions:

We have

and also

Thus, the multivariate Taylor series expansion of about the point is

We can also write this in coordinate-independent form as

9.3.2 Nature of Stationary Points and the Hessian

Suppose is a stationary point with

Around :

where .

• If is positive definite at , then for all near , so is a local minimum.

• If is negative definite at , then for all near , so is a local maximum.

• If is indefinite at , then it may be a maximum, minimum or saddle point.

9.3.2.1 Definiteness and Eigenvalues

If is a real symmetric matrix, then we can diagonalise it by an orthogonal transformation (by results in IA Vectors and Matrices). Using coordinates along the principal axes (eigenvectors), in dimensions:

Hence,

• is positive definite iff all eigenvalues (minimum),

• is negative definite iff all eigenvalues (maximum),

• If all eigenvalues are non-zero, but are of mixed signs, then this corresponds to a saddle point.

• If any of the eigenvalues are zero, then we need higher order terms in the Taylor series to classify the stationary point.

9.3.2.2 Definiteness and Signature

An alternative method to determine definiteness without having to compute eigenvalues is to use signatures.

9.3.3 Contours Near Stationary Points

Suppose has a stationary point at . Using coordinates aligned with the principal axes of the Hessian matrix at , we have

Assume that the eigenvalues are non-zero. Then, consider

then around we have

On contours near , since is constant, we have

• At a maximum or minimum, and have the same sign, and the contours are ellipses.

• At a saddle point, and have opposite signs, and the contours are hyperbolae.

9.4 Systems of Linear ODEs

Consider and with

where , , , are constants. We can write this in vector form as

where

There are two ways to solve this system:

1. Convert to a single higher order ODE for one variable.

   We have

   Now we have a linear 2nd order ODE with constant coefficients.

2. Solve directly with matrix methods. [This may be more convenient.]

   Remark. Under some cases, we write higher order ODE as a set of 1st order ODEs, esentially reversing the process above.

   Example 9.10

   Consider the equation

   We can let , and . We then have

   Hence,

9.4.1 Matrix methods

Consider

where is a constant matrix.

1. Write where is the complementary function satisfying , and is a particular integral.

2. Look for of the form where is a constant vector. Then,

   Since holds for all , taking we have

   Hence is an eigenvalue of , and is the corresponding eigenvector.

   For a system of equations, we have such complementary functions if eigenvalues are distinct.

3. Find a that satisfies by trying an appropriate form.

9.4.2 Non-Degenerate Phase portraits

For autonomous systems, there is one trajectory through each point in phase space, except at fixed points.

Consider the homogeneous equation

There is a fixed point at . For , the general solution for (non-degenerate case) is

where are constants.

For and , we have the following cases:

1. and are real and of opposite signs. WLOG suppose . In this case, can be chosen to be real. The fixed point is a saddle node.

   

2. and are real and have the same sign. WLOG suppose .

   • If both are positive, then the fixed point is an unstable node.
   
   • If both are negative, then the fixed point is a stable node.
   

3. and are complex conjugates, then and . Then,

   where .

   • If , then we have a unstable spiral.
   
   • If , then we have a stable spiral.
   
   • If , then we have a centre, with closed elliptical trajectories.
   

In order to determine the direction of motion along the trajectories, we can evaluate at some points on the trajectory.

For example, if at , then motion is upwards at that point, so the direction of motion is counter-clockwise.

9.5 Non-Linear Dynamical Systems

We aim to use techniques for linear systems to investigate the nature of equilibrium points.

Consider an autonomous system of 2 non-linear first-order ODEs:

where and are general non-linear functions of and . [They do not depend explicitly on .]

An equilibrium (fixed) point of the system is a point at which and , i.e.

We need to solve simultaneously to determine the fixed points.

Stabilities of the fixed points can be deduced from perturbation analysis:

where and are small perturbations. We have

We can write this in matrix form as

This is a linear system of homogenous ODEs, and hence the eigenvalues of determine the stability of the fixed point.

9.6 Partial Differential Equations

Partial differential equations (PDEs) involve several independent variables. We will illustrate some ideas with wave equations.

9.6.1 First-Order Wave Equation

Consider , where is the spatial coordinate and is time, with

where is a constant with dimensions of velocity.

We can solve this by the method of characteristics. We consider how vary along a path , so that we consider . We have

If we choose such that , then where is a constant, and we have

Paths where are called characteristics of .

Since is constant along characteristics, the general solution of is

where is an arbitrary function.

This expression translates the -dependence of at to the left by at time .

The solutions are left-moving wave solutions.

9.6.2 Second-Order Wave Equation

A lot of physical systems allow waves to propagate in both directions. This is modelled by second-order wave equations.

Consider

where is a constant with dimensions of velocity.

Since the differential operator can be factorised as

we can write the wave equation as

These two operators commute, so both and are solutions, where and are arbitrary functions.

This suggests that the general solution is

where and are arbitrary functions.