What is the solution for (d^2y)/(dx^2)-10(dy)/(dx)+5^2y=0 ?

The solution for (d^2y)/(dx^2)-10(dy)/(dx)+5^2y=0 is y=c_{1}e^{5x}+c_{2}xe^{5x}

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(d^2y)/(dx^2)-10(dy)/(dx)+5^2y=0

\frac{d ^{2} y}{d x ^{2}} - 10 \frac{d y}{d x} + 5^{2} y = 0

y = c_{1} e^{5 x} + c_{2} x e^{5 x}

Solution steps

\frac{d ^{2} y}{d x ^{2}} - 10 \frac{d y}{d x} + 5^{2} y = 0

Solve linear ODE:

y = c_{1} e^{5 x} + c_{2} x e^{5 x}

y = c_{1} e^{5 x} + c_{2} x e^{5 x}

12 \frac{d ^{2} y}{d x ^{2}} - 5 \frac{d y}{d x} - 3 y = 0

\frac{d ^{2} y}{d x ^{2}} (1) - \frac{d y}{d x} (10) = 0

y^{^{'''}} - 15 y^{^{''}} + 72 y^{^{'}} - 108 y = 0

y^{^{'''}} + (- 7) y^{^{''}} + 8 y^{^{'}} + 16 y = 0

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

What is the solution for (d^2y)/(dx^2)-10(dy)/(dx)+5^2y=0 ?
The solution for (d^2y)/(dx^2)-10(dy)/(dx)+5^2y=0 is y=c_{1}e^{5x}+c_{2}xe^{5x}