What is the solution for y^{'''}-2y^{''}+5y^'=0,y(0)=0,y^'(0)=0 ?

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

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

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

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

Solution steps

y^{'''} (t) - 2 y^{''} (t) + 5 y^{'} (t) = 0

Solve linear ODE:

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

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

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