Question

Is this equation solvable? And how?

y” = Ay + B

A and B are (real) constants. I tried doing undetermined coefficients but it didn’t work out for me. The homogeneous part is easy enough.

Thanks.

Answer 1

You can start by assuming that your solution has the form

    y(x) = M*exp(sqrt(A)*x) + N*exp(-sqrt(A)*x) + C

since the exponential parts are the solution to the homogeneous equation.

Now we can substitute this back into our differential equation and try to solve for C.

    y" = M*A*exp(sqrt(A)*x) + N*A*exp(-sqrt(A)*x)
    M*A*exp(sqrt(A)*x) + N*A*exp(-sqrt(A)*x) = 
        A*(M*exp(sqrt(A)*x) + N*exp(-sqrt(A)*x) + C) + B
    0 = A*C + B
    C = -B/A.

Therefore:

    y = M*exp(sqrt(A)*x) + N*exp(-sqrt(A)*x) - B/A.

This example worked because it’s just a constant being added to our equation, other inhomogeneous differential equations can however still be solved using Green’s function, once you have the solution to the corresponding homogeneous equation.