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Popular >

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

Solution

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

Solution

y = e^{- \frac{x}{2}} (c_{1} cos (\frac{3 x}{2}) + c_{2} sin (\frac{3 x}{2})) + e^{- \frac{x}{2}} cos (\frac{3 x}{2}) \cdot \int - \frac{2 3 e ^{\frac{x}{2}} sin ^{2} ( x ) sin ( \frac{3 x}{2} )}{3} d x + e^{- \frac{x}{2}} sin (\frac{3 x}{2}) \cdot \int \frac{2 3 e ^{\frac{x}{2}} sin ^{2} ( x ) cos ( \frac{3 x}{2} )}{3} d x

Solution steps

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

Solve linear ODE:

y = e^{- \frac{x}{2}} (c_{1} cos (\frac{3 x}{2}) + c_{2} sin (\frac{3 x}{2})) + e^{- \frac{x}{2}} cos (\frac{3 x}{2}) \cdot \int - \frac{2 3 e ^{\frac{x}{2}} sin ^{2} ( x ) sin ( \frac{3 x}{2} )}{3} d x + e^{- \frac{x}{2}} sin (\frac{3 x}{2}) \cdot \int \frac{2 3 e ^{\frac{x}{2}} sin ^{2} ( x ) cos ( \frac{3 x}{2} )}{3} d x

y = e^{- \frac{x}{2}} (c_{1} cos (\frac{3 x}{2}) + c_{2} sin (\frac{3 x}{2})) + e^{- \frac{x}{2}} cos (\frac{3 x}{2}) \cdot \int - \frac{2 3 e ^{\frac{x}{2}} sin ^{2} ( x ) sin ( \frac{3 x}{2} )}{3} d x + e^{- \frac{x}{2}} sin (\frac{3 x}{2}) \cdot \int \frac{2 3 e ^{\frac{x}{2}} sin ^{2} ( x ) cos ( \frac{3 x}{2} )}{3} d x

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Frequently Asked Questions (FAQ)

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