What is the solution for (d^2y)/(dx^2)-6(dy)/(dx)+8y=4e^{3x}+8x ?

The solution for (d^2y)/(dx^2)-6(dy)/(dx)+8y=4e^{3x}+8x is y=c_{1}e^{4x}+c_{2}e^{2x}-4e^{3x}+x+3/4

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(d^2y)/(dx^2)-6(dy)/(dx)+8y=4e^{3x}+8x

\frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} + 8 y = 4 e^{3 x} + 8 x

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

Solution steps

\frac{d ^{2} y}{d x ^{2}} - 6 \frac{d y}{d x} + 8 y = 4 e^{3 x} + 8 x

Solve linear ODE:

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

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

What is the solution for (d^2y)/(dx^2)-6(dy)/(dx)+8y=4e^{3x}+8x ?
The solution for (d^2y)/(dx^2)-6(dy)/(dx)+8y=4e^{3x}+8x is y=c_{1}e^{4x}+c_{2}e^{2x}-4e^{3x}+x+3/4