en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

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

Solution

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

Popular Examples

y^{''}-2y^'-3y=4,y(0)=1,y^'(0)=-1

y^{^{''}} - 2 y^{^{'}} - 3 y = 4, y (0) = 1, y^{^{'}} (0) = - 1

y^{''}+5y^'+6y=20sin(2t)

y^{^{''}} + 5 y^{^{'}} + 6 y = 20 sin (2 t)

y^{''}-2y^'+y=25sin(2x)

y^{^{''}} - 2 y^{^{'}} + y = 25 sin (2 x)

y^{''}-8y^'+20y=200x^2-65xe^x

y^{^{''}} - 8 y^{^{'}} + 20 y = 200 x^{2} - 65 x e^{x}

y^{''}-4y^'+4y=e^t

y^{^{''}} - 4 y^{^{'}} + 4 y = e^{t}

Frequently Asked Questions (FAQ)

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