What is the solution for (d^4y)/(dx^4)-2((d^2y)/(dx^2))+y=0 ?

The solution for (d^4y)/(dx^4)-2((d^2y)/(dx^2))+y=0 is y=c_{1}e^x+c_{2}xe^x+c_{3}e^{-x}+c_{4}xe^{-x}

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(d^4y)/(dx^4)-2((d^2y)/(dx^2))+y=0

\frac{d ^{4} y}{d x ^{4}} - 2 (\frac{d ^{2} y}{d x ^{2}}) + y = 0

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

Solution steps

\frac{d ^{4} y}{d x ^{4}} - 2 (\frac{d ^{2} y}{d x ^{2}}) + y = 0

Solve linear ODE:

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

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

y^{^{''}} - 16 y = 0, y (0) = 0, y^{^{'}} (0) = 1

y^{^{''}} - 7 y^{^{'}} = 0, y (0) = 2 - 2 e^{7}, y (1) = 0

\frac{d ^{2} y}{d x ^{2}} + 15 \frac{d y}{d x} + 56 y = 0

y^{^{''}} + 4 y^{^{'}} + k y = 0

y^{^{''}} + 4 y = 0, y (0) = 6, y^{^{'}} (0) = - 6

What is the solution for (d^4y)/(dx^4)-2((d^2y)/(dx^2))+y=0 ?
The solution for (d^4y)/(dx^4)-2((d^2y)/(dx^2))+y=0 is y=c_{1}e^x+c_{2}xe^x+c_{3}e^{-x}+c_{4}xe^{-x}