en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

y^{''}+4y=16t^3sin(2t)

Solution

y^{^{''}} + 4 y = 16 t^{3} sin (2 t)

Solution

y = c_{1} cos (2 t) + c_{2} sin (2 t) - 8 cos (2 t) (\frac{1}{4} t^{4} sin^{2} (2 t) - \frac{1}{256} (- 32 t^{4} cos (4 t) + 32 t^{3} sin (4 t) + 24 t^{2} cos (4 t) - 12 t sin (4 t) - 3 cos (4 t))) + sin (2 t) (- t^{3} cos (4 t) + \frac{3}{4} t^{2} sin (4 t) + \frac{3}{8} t cos (4 t) - \frac{3}{32} sin (4 t))

Solution steps

y^{''} (t) + 4 y = 16 t^{3} sin (2 t)

Solve linear ODE:

y = c_{1} cos (2 t) + c_{2} sin (2 t) - 8 cos (2 t) (\frac{1}{4} t^{4} sin^{2} (2 t) - \frac{1}{256} (- 32 t^{4} cos (4 t) + 32 t^{3} sin (4 t) + 24 t^{2} cos (4 t) - 12 t sin (4 t) - 3 cos (4 t))) + sin (2 t) (- t^{3} cos (4 t) + \frac{3}{4} t^{2} sin (4 t) + \frac{3}{8} t cos (4 t) - \frac{3}{32} sin (4 t))

y = c_{1} cos (2 t) + c_{2} sin (2 t) - 8 cos (2 t) (\frac{1}{4} t^{4} sin^{2} (2 t) - \frac{1}{256} (- 32 t^{4} cos (4 t) + 32 t^{3} sin (4 t) + 24 t^{2} cos (4 t) - 12 t sin (4 t) - 3 cos (4 t))) + sin (2 t) (- t^{3} cos (4 t) + \frac{3}{4} t^{2} sin (4 t) + \frac{3}{8} t cos (4 t) - \frac{3}{32} sin (4 t))

Graph

Popular Examples

y^{''}+11y^'+18y=5

y^{^{''}} + 11 y^{^{'}} + 18 y = 5

y^{''}-y^'+14y=2+e^{x/2}

y^{^{''}} - y^{^{'}} + 14 y = 2 + e^{\frac{x}{2}}

y^{''}+4y=t^2sin(2t)+(6t+7)cos(2t)

y^{^{''}} + 4 y = t^{2} sin (2 t) + (6 t + 7) cos (2 t)

x^{''}+25x=t^2+t

x^{^{''}} + 25 x = t^{2} + t

y^{''}+4y=cos(x),y(0)=6,y^'(0)=6

y^{^{''}} + 4 y = cos (x), y (0) = 6, y^{^{'}} (0) = 6

Frequently Asked Questions (FAQ)

What is the solution for y^{''}+4y=16t^3sin(2t) ?
The solution for y^{''}+4y=16t^3sin(2t) is