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Popular >

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

Solution

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

Solution

y = c_{1} cos (2 x) + c_{2} sin (2 x) + (- \frac{1}{4} x sin^{4} (x) - \frac{1}{16} sin^{3} (x) cos (x) + \frac{3}{32} x - \frac{3}{64} sin (2 x)) cos (2 x) + \frac{1}{128} (- 8 x^{2} + 32 x sin^{2} (x) sin (2 x) + 4 x sin (4 x) - 16 sin^{4} (x) + cos (4 x)) sin (2 x)

Solution steps

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

Solve linear ODE:

y = c_{1} cos (2 x) + c_{2} sin (2 x) + (- \frac{1}{4} x sin^{4} (x) - \frac{1}{16} sin^{3} (x) cos (x) + \frac{3}{32} x - \frac{3}{64} sin (2 x)) cos (2 x) + \frac{1}{128} (- 8 x^{2} + 32 x sin^{2} (x) sin (2 x) + 4 x sin (4 x) - 16 sin^{4} (x) + cos (4 x)) sin (2 x)

y = c_{1} cos (2 x) + c_{2} sin (2 x) + (- \frac{1}{4} x sin^{4} (x) - \frac{1}{16} sin^{3} (x) cos (x) + \frac{3}{32} x - \frac{3}{64} sin (2 x)) cos (2 x) + \frac{1}{128} (- 8 x^{2} + 32 x sin^{2} (x) sin (2 x) + 4 x sin (4 x) - 16 sin^{4} (x) + cos (4 x)) sin (2 x)

Graph

Popular Examples

y^{''}+9y=cos(t),y(0)=0,y^'(0)=0

y^{^{''}} + 9 y = cos (t), y (0) = 0, y^{^{'}} (0) = 0

y^{''}-y=sin(2t)

y^{^{''}} - y = sin (2 t)

y^{^{''}} + 6 y^{^{'}} = 2

y^{^{''}} + 6 y^{^{'}} = 1

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

y^{^{''}} + 2 y^{^{'}} + 2 y = \frac{( e ^{- x} )}{cos ( x )}

Frequently Asked Questions (FAQ)

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