What is the solution for y^{''}-4y^'+5y=12e^t,y(0)=6,y^'(0)=7 ?

The solution for y^{''}-4y^'+5y=12e^t,y(0)=6,y^'(0)=7 is y=e^{2t}sin(t)+6e^t

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y^{''}-4y^'+5y=12e^t,y(0)=6,y^'(0)=7

y^{^{''}} - 4 y^{^{'}} + 5 y = 12 e^{t}, y (0) = 6, y^{^{'}} (0) = 7

y = e^{2 t} sin (t) + 6 e^{t}

Solution steps

y^{''} (t) - 4 y^{'} (t) + 5 y = 12 e^{t}

Solve linear ODE:

y = e^{2 t} sin (t) + 6 e^{t}

y = e^{2 t} sin (t) + 6 e^{t}

(D^{2} - D - 2) y = 3 e^{2 x}

\frac{d ^{2} y}{d t ^{2}} - 3 \frac{d y}{d t} - 18 y = (t + 1) e^{t}

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

\frac{d ^{2} y}{d x ^{2}} + 4 y = cos (2 x) + sin (2 x)

y^{^{''}} - 2 y^{^{'}} + 5 y = 104 sin (3 x)

What is the solution for y^{''}-4y^'+5y=12e^t,y(0)=6,y^'(0)=7 ?
The solution for y^{''}-4y^'+5y=12e^t,y(0)=6,y^'(0)=7 is y=e^{2t}sin(t)+6e^t