What is the solution for (d^2y)/(dx^2)+3(dy)/(dx)-4y=x^2 ?

The solution for (d^2y)/(dx^2)+3(dy)/(dx)-4y=x^2 is y=c_{1}e^x+c_{2}e^{-4x}-(x^2)/4-(3x)/8-13/32

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(d^2y)/(dx^2)+3(dy)/(dx)-4y=x^2

\frac{d ^{2} y}{d x ^{2}} + 3 \frac{d y}{d x} - 4 y = x^{2}

y = c_{1} e^{x} + c_{2} e^{- 4 x} - \frac{x ^{2}}{4} - \frac{3 x}{8} - \frac{13}{32}

Solution steps

\frac{d ^{2} y}{d x ^{2}} + 3 \frac{d y}{d x} - 4 y = x^{2}

Solve linear ODE:

y = c_{1} e^{x} + c_{2} e^{- 4 x} - \frac{x ^{2}}{4} - \frac{3 x}{8} - \frac{13}{32}

y = c_{1} e^{x} + c_{2} e^{- 4 x} - \frac{x ^{2}}{4} - \frac{3 x}{8} - \frac{13}{32}

y^{^{''}} + 3 y^{^{'}} + 2 y = - 6 - 4 x - 7 x^{2} - 2 x^{3}

(D^{2} + 1) y = 12 cos^{2} (x)

(D^{2} + 3 D + 2) y = 1 + 3 x + x^{2}

y^{^{''}} - 4 y = cos (x) - sin (x)

y^{^{''}} + 8 y^{^{'}} + 12 y = x e^{2 x}

What is the solution for (d^2y)/(dx^2)+3(dy)/(dx)-4y=x^2 ?
The solution for (d^2y)/(dx^2)+3(dy)/(dx)-4y=x^2 is y=c_{1}e^x+c_{2}e^{-4x}-(x^2)/4-(3x)/8-13/32