differential equations - 2nd order ODE undetermined coefficients -

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is equation solvable? , how?

y" = ay + b

a , b (real) constants. tried doing undetermined coefficients didn't work out me. homogeneous part easy enough.

thanks.

you can start assuming solution has form

    y(x) = m*exp(sqrt(a)*x) + n*exp(-sqrt(a)*x) + c

since exponential parts solution homogeneous equation.

now can substitute our differential equation , try solve 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 constant being added our equation, other inhomogeneous differential equations can still solved using green's function, once have solution corresponding homogeneous equation.


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