What is the solution for x^{''}+a^2x=a^2b+c ?

The solution for x^{''}+a^2x=a^2b+c is x=c_{1}cos(at)+c_{2}sin(at)+(a^2b+c)/(a^2)

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x^{''}+a^2x=a^2b+c

x^{^{''}} + a^{2} x = a^{2} b + c

x = c_{1} cos (a t) + c_{2} sin (a t) + \frac{a ^{2} b + c}{a ^{2}}

Solution steps

x^{''} (t) + a^{2} x = a^{2} b + c

Solve linear ODE:

x = c_{1} cos (a t) + c_{2} sin (a t) + \frac{a ^{2} b + c}{a ^{2}}

x = c_{1} cos (a t) + c_{2} sin (a t) + \frac{a ^{2} b + c}{a ^{2}}

y^{^{''}} - 10 y^{^{'}} + 9 y = 5

y^{^{''}} - 8 y^{^{'}} + 20 y = 200 x^{2} - 78 x e^{x}

y^{^{''}} + y^{^{'}} - 12 y = 2 cos (t)

D^{2} (D - 1) y = 8 e^{2 x}

x^{^{''}} - x^{^{'}} - 6 x = 2 e^{4 t}

What is the solution for x^{''}+a^2x=a^2b+c ?
The solution for x^{''}+a^2x=a^2b+c is x=c_{1}cos(at)+c_{2}sin(at)+(a^2b+c)/(a^2)