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Popular >

2y^{''}+2y^'+y=4sqrt(x)

Solution

2 y^{^{''}} + 2 y^{^{'}} + y = 4 x

Solution

y = e^{- \frac{x}{2}} (c_{1} cos (\frac{x}{2}) + c_{2} sin (\frac{x}{2})) + e^{- \frac{x}{2}} cos (\frac{x}{2}) \cdot \int - 4 e^{\frac{x}{2}} x sin (\frac{x}{2}) d x + e^{- \frac{x}{2}} sin (\frac{x}{2}) \cdot \int 4 e^{\frac{x}{2}} x cos (\frac{x}{2}) d x

Solution steps

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

Solve linear ODE:

y = e^{- \frac{x}{2}} (c_{1} cos (\frac{x}{2}) + c_{2} sin (\frac{x}{2})) + e^{- \frac{x}{2}} cos (\frac{x}{2}) \cdot \int - 4 e^{\frac{x}{2}} x sin (\frac{x}{2}) d x + e^{- \frac{x}{2}} sin (\frac{x}{2}) \cdot \int 4 e^{\frac{x}{2}} x cos (\frac{x}{2}) d x

y = e^{- \frac{x}{2}} (c_{1} cos (\frac{x}{2}) + c_{2} sin (\frac{x}{2})) + e^{- \frac{x}{2}} cos (\frac{x}{2}) \cdot \int - 4 e^{\frac{x}{2}} x sin (\frac{x}{2}) d x + e^{- \frac{x}{2}} sin (\frac{x}{2}) \cdot \int 4 e^{\frac{x}{2}} x cos (\frac{x}{2}) d x

Popular Examples

y^{''}+y^'=sin(x)

y^{^{''}} + y^{^{'}} = sin (x)

y^{''}+2y^'+5y=4e^{-x}cos(2x)

y^{^{''}} + 2 y^{^{'}} + 5 y = 4 e^{- x} cos (2 x)

y^{''}-y^'-2y=2e^{-x}

y^{^{''}} - y^{^{'}} - 2 y = 2 e^{- x}

y^{''}+5y^'-6y=xe^{-5x}

y^{^{''}} + 5 y^{^{'}} - 6 y = x e^{- 5 x}

y^{''}+7y^'+12y=t^2e^{-3t}

y^{^{''}} + 7 y^{^{'}} + 12 y = t^{2} e^{- 3 t}

Frequently Asked Questions (FAQ)

What is the solution for 2y^{''}+2y^'+y=4sqrt(x) ?
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