You have:Originally Posted bykingkaisai2

with initial condition when .

Now rewrite the DE as:

and we see that we have an inhomogeneous linear ODE with constant

coefficients, and so a general solution is the sum of the general solution

of the homogeneous equation:

,

and any particular integral of the original equation (1).

Now the homogeneous equation (2) can be solved by a number of methods,

we can use a trial solution , substitute this into

the equation and find the value of that gives the correct answer.

Another method is to note that (2) is of variable separable type.

I will use the first of these methods, I will suppose that: , then:

,

so if this is a solution we must have , and the general

solution of (2) is: .

Now we need to find a particular integral for (1). Here we note that the RHS is

a multiple of , so if the LHS can be made to equal

the RHS with suitable choice of and , so substituting this into (1) we get:

.

So equating the coefficient of in the above gives , so ,

and the constant terms gives , or .

Hence a particular integral of (1) is:

.

Combining this with the general solution of (2) found earlier we get the

general solution of (1) is:

.

Now we need to fit the initial conditions. Putting into (3)

we have:

which we solve to find that is applicable for this problem.

RonL