What is the solution for (d^2+2)y=x^2e^3x ?

The solution for (d^2+2)y=x^2e^3x is y=c_{1}cos(sqrt(2)x)+c_{2}sin(sqrt(2)x)+(e^3x^3)/2-(3e^3x)/2

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(d^2+2)y=x^2e^3x

(d^{2} + 2) y = x^{2} e^{3} x

y = c_{1} cos (2 x) + c_{2} sin (2 x) + \frac{e ^{3} x ^{3}}{2} - \frac{3 e ^{3} x}{2}

Solution steps

(d^{2} + 2) y = x^{2} e^{3} x

Solve linear ODE:

y = c_{1} cos (2 x) + c_{2} sin (2 x) + \frac{e ^{3} x ^{3}}{2} - \frac{3 e ^{3} x}{2}

y = c_{1} cos (2 x) + c_{2} sin (2 x) + \frac{e ^{3} x ^{3}}{2} - \frac{3 e ^{3} x}{2}

y^{^{''}} - 4 y^{^{'}} + 4 y = (x + 1) e^{2} x

(d^{2} - 4 d - 12) \cdot y = (x - 1) \cdot e^{- 2 x}

(d^{2} + 4 d + 4) y = x^{2} + e^{x} + cos (x)

(d^{2} - 3 d + 2) y = 5 e^{3} x

(d^{2} - 4 d + 4) y = 8 x^{2} e^{2} x

What is the solution for (d^2+2)y=x^2e^3x ?
The solution for (d^2+2)y=x^2e^3x is y=c_{1}cos(sqrt(2)x)+c_{2}sin(sqrt(2)x)+(e^3x^3)/2-(3e^3x)/2