What is the solution for y^{''}-a^2*y+b=0 ?

The solution for y^{''}-a^2*y+b=0 is y=c_{1}e^{at}+c_{2}e^{-at}+b/(a^2)

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y^{''}-a^2*y+b=0

y^{^{''}} - a^{2} \cdot y + b = 0

y = c_{1} e^{a t} + c_{2} e^{- a t} + \frac{b}{a ^{2}}

Solution steps

y^{''} (t) - a^{2} \cdot y + b = 0

Solve linear ODE:

y = c_{1} e^{a t} + c_{2} e^{- a t} + \frac{b}{a ^{2}}

y = c_{1} e^{a t} + c_{2} e^{- a t} + \frac{b}{a ^{2}}

y^{^{''}} - 4 y = 8 e^{2 x}, y (0) = 1, y (0) = 0

y^{^{''}} + 4 y^{^{'}} = 2 e^{- 4 x} + 3 sin (2 x)

y^{^{''}} - 3 y^{^{'}} + 2 y = cos (t), y (0) = 0, y^{^{'}} (0) = 1

y^{^{''}} - 2 y^{^{'}} - 3 y = 4 e^{2 x} + 2 x^{3}

\frac{d ^{2} y}{d x ^{2}} + 10 \frac{d y}{d x} + 16 y = 40 cos (x)

What is the solution for y^{''}-a^2*y+b=0 ?
The solution for y^{''}-a^2*y+b=0 is y=c_{1}e^{at}+c_{2}e^{-at}+b/(a^2)