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Popular >

y^{''}+2y^'+y=t^5e^{-5t}

Solution

y^{^{''}} + 2 y^{^{'}} + y = t^{5} e^{- 5 t}

Solution

y = c_{1} e^{- t} + c_{2} t e^{- t} + \frac{e ^{- 5 t} t ^{5}}{16} + \frac{5 e ^{- 5 t} t ^{4}}{32} + \frac{15 e ^{- 5 t} t ^{3}}{64} + \frac{15 e ^{- 5 t} t ^{2}}{64} + \frac{75 e ^{- 5 t} t}{512} + \frac{45}{1024} e^{- 5 t}

Solution steps

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

Solve linear ODE:

y = c_{1} e^{- t} + c_{2} t e^{- t} + \frac{e ^{- 5 t} t ^{5}}{16} + \frac{5 e ^{- 5 t} t ^{4}}{32} + \frac{15 e ^{- 5 t} t ^{3}}{64} + \frac{15 e ^{- 5 t} t ^{2}}{64} + \frac{75 e ^{- 5 t} t}{512} + \frac{45}{1024} e^{- 5 t}

y = c_{1} e^{- t} + c_{2} t e^{- t} + \frac{e ^{- 5 t} t ^{5}}{16} + \frac{5 e ^{- 5 t} t ^{4}}{32} + \frac{15 e ^{- 5 t} t ^{3}}{64} + \frac{15 e ^{- 5 t} t ^{2}}{64} + \frac{75 e ^{- 5 t} t}{512} + \frac{45}{1024} e^{- 5 t}

Popular Examples

y^{''}+16y=9(t)-9(t-98pi),y(0)=0,y^'(0)=0

y^{^{''}} + 16 y = 9 (t) - 9 (t - 98 π), y (0) = 0, y^{^{'}} (0) = 0

y^{''}-4y^'+3y=8e^{-x}+9

y^{^{''}} - 4 y^{^{'}} + 3 y = 8 e^{- x} + 9

y^{''}+18y^'+82y=2e^{-9x}cos(x)

y^{^{''}} + 18 y^{^{'}} + 82 y = 2 e^{- 9 x} cos (x)

y^{''}+9y^'=t^2e^{3t}+6

y^{^{''}} + 9 y^{^{'}} = t^{2} e^{3 t} + 6

(d^2y)/(dt^2)-12(dy)/(dt)+36y=7+4t

\frac{d ^{2} y}{d t ^{2}} - 12 \frac{d y}{d t} + 36 y = 7 + 4 t

Frequently Asked Questions (FAQ)

What is the solution for y^{''}+2y^'+y=t^5e^{-5t} ?
The solution for y^{''}+2y^'+y=t^5e^{-5t} is