Consider

A particular integral is , since substituting it gives

Then the complementary function is the solution of the homogeneous equation , which we have already solved as .

Hence, the general solution is

Consider the decay between three isotopes , with decay constants for and respectively.

Thus we have

and also

We shall try the particular integral of the form . Substituting it gives

Hence is the solution of the homogeneous equation . Thus

Thus, the general solution is

Now, if we were given the initial conditions , then

Hence,

We have

We can identify and . Thus, the integrating factor is

Hence,

Let us consider two cases.

• If , then

  This is exactly the solution we arrived at in Example 4.15, using the PI and CF method.

• If , then

  Thus,

  Note that the particular integral is now proportional to , which is different from the previous case. This is called the resonance case.