You have:Originally Posted by kingkaisai2
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)
which we solve to find that is applicable for this problem.