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Popular Calculus >

y^{''}+2y^'+5y=5\H(t)

Solution

y^{^{''}} + 2 y^{^{'}} + 5 y = 5 H (t)

Solution

y = e^{- t} (c_{1} cos (2 t) + c_{2} sin (2 t)) + e^{- t} cos (2 t) \cdot \int - \frac{5 e ^{t} sin ( 2 t ) H ( t )}{2} d t + e^{- t} sin (2 t) \cdot \int \frac{5 e ^{t} cos ( 2 t ) H ( t )}{2} d t

Solution steps

y^{''} (t) + 2 y^{'} (t) + 5 y = 5 H (t)

Solve linear ODE:

y = e^{- t} (c_{1} cos (2 t) + c_{2} sin (2 t)) + e^{- t} cos (2 t) \cdot \int - \frac{5 e ^{t} sin ( 2 t ) H ( t )}{2} d t + e^{- t} sin (2 t) \cdot \int \frac{5 e ^{t} cos ( 2 t ) H ( t )}{2} d t

y = e^{- t} (c_{1} cos (2 t) + c_{2} sin (2 t)) + e^{- t} cos (2 t) \cdot \int - \frac{5 e ^{t} sin ( 2 t ) H ( t )}{2} d t + e^{- t} sin (2 t) \cdot \int \frac{5 e ^{t} cos ( 2 t ) H ( t )}{2} d t

Popular Examples

y^{''}+3*y^'+2*y=sqrt(x)

y^{''}-2y^'-3y=e^{3x}

(d^2x)/(dt^2)+(dx)/(dt)+x=1,x(0)=0

y^{''}-y^'+y=14sin(t)

(d^2y)/(dx^2)-5(dy)/(dx)-14y=(x+1)e^x

Frequently Asked Questions (FAQ)

What is the solution for y^{''}+2y^'+5y=5\H(t) ?
The solution for y^{''}+2y^'+5y=5\H(t) is