What is the solution for y^{''''}-4y^{'''}+5y^{''}=0 ?

The solution for y^{''''}-4y^{'''}+5y^{''}=0 is y=c_{1}+c_{2}t+e^{2t}(c_{3}cos(t)+c_{4}sin(t))

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y^{''''}-4y^{'''}+5y^{''}=0

y^{^{''''}} - 4 y^{^{'''}} + 5 y^{^{''}} = 0

y = c_{1} + c_{2} t + e^{2 t} (c_{3} cos (t) + c_{4} sin (t))

Solution steps

y^{(4)} (t) - 4 y^{'''} (t) + 5 y^{''} (t) = 0

Solve linear ODE:

y = c_{1} + c_{2} t + e^{2 t} (c_{3} cos (t) + c_{4} sin (t))

y = c_{1} + c_{2} t + e^{2 t} (c_{3} cos (t) + c_{4} sin (t))

y^{^{'''}} + 4 y = 0

y^{^{''}} + 4 y^{^{'}} + 2.5 y = 0

y^{^{'''}} - 3 y^{^{'}} - 2 y = 0

(d^{3} + d^{2} + 4 d + 4) \cdot y = 0

y^{^{''}} - 2 y^{^{'}} - 3 y = 0, y (0) = 5, y^{^{'}} (0) = - 1

What is the solution for y^{''''}-4y^{'''}+5y^{''}=0 ?
The solution for y^{''''}-4y^{'''}+5y^{''}=0 is y=c_{1}+c_{2}t+e^{2t}(c_{3}cos(t)+c_{4}sin(t))