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Popular Calculus >

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

Solution

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

Solution

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + e^{2 x} \cdot \int - \frac{4 x sin ^{2} ( x )}{e ^{2 x}} d x + 4 e^{2 x} x (- \frac{1}{2} e^{- 2 x} sin^{2} (x) + \frac{1}{8} (- e^{- 2 x} cos (2 x) - e^{- 2 x} sin (2 x)))

Solution steps

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

Solve linear ODE:

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + e^{2 x} \cdot \int - \frac{4 x sin ^{2} ( x )}{e ^{2 x}} d x + 4 e^{2 x} x (- \frac{1}{2} e^{- 2 x} sin^{2} (x) + \frac{1}{8} (- e^{- 2 x} cos (2 x) - e^{- 2 x} sin (2 x)))

y = c_{1} e^{2 x} + c_{2} x e^{2 x} + e^{2 x} \cdot \int - \frac{4 x sin ^{2} ( x )}{e ^{2 x}} d x + 4 e^{2 x} x (- \frac{1}{2} e^{- 2 x} sin^{2} (x) + \frac{1}{8} (- e^{- 2 x} cos (2 x) - e^{- 2 x} sin (2 x)))

Popular Examples

y^{''}-6y^'+30y=15sin(x)

y^{''}+2y^'+10y=25^2+3

y^{''}-2y^'+2y=4x+5,y(0)=3,y^'(0)=0

y^{''}+1y=(sec(x))tan(x)

y^{''}-y^'-2y=13x+9

Frequently Asked Questions (FAQ)

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