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Popular Calculus >

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

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

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

y^{''}-2y^'+2y=e^xtan(x)

Frequently Asked Questions (FAQ)

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