What is the solution for a^2y^{''}-(a^2+1)y^'+y=x ?

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

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a^2y^{''}-(a^2+1)y^'+y=x

a^{2} y^{^{''}} - (a^{2} + 1) y^{^{'}} + y = x

y = c_{1} e^{x} + c_{2} e^{\frac{x}{a ^{2}}} + x + a^{2} + 1

Solution steps

a^{2} y^{''} (x) - (a^{2} + 1) y^{'} (x) + y = x

Solve linear ODE:

y = c_{1} e^{x} + c_{2} e^{\frac{x}{a ^{2}}} + x + a^{2} + 1

y = c_{1} e^{x} + c_{2} e^{\frac{x}{a ^{2}}} + x + a^{2} + 1

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} - 12 y = 3 e^{5 x}

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d y}{d x} - 12 y = cos (x)

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

\frac{d ^{2} y}{d x ^{2}} - 16 y = 9 e^{4 x} + 30 e^{x}

y^{^{''}} - 5 y^{^{'}} + 6 y^{^{'}} = x^{2}

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