What is the solution for y^{''}-12y^'+45y=0,y(0)=5,y^'(0)=-3 ?

The solution for y^{''}-12y^'+45y=0,y(0)=5,y^'(0)=-3 is y=e^{6t}(5cos(3t)-11sin(3t))

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y^{''}-12y^'+45y=0,y(0)=5,y^'(0)=-3

y^{^{''}} - 12 y^{^{'}} + 45 y = 0, y (0) = 5, y^{^{'}} (0) = - 3

y = e^{6 t} (5 cos (3 t) - 11 sin (3 t))

Solution steps

y^{''} (t) - 12 y^{'} (t) + 45 y = 0

Solve linear ODE:

y = e^{6 t} (5 cos (3 t) - 11 sin (3 t))

y = e^{6 t} (5 cos (3 t) - 11 sin (3 t))

10 y^{^{''}} + 45 y^{^{'}} + 2500 y = 0

y^{^{''}} + 4 y^{^{'}} + 4 y = 0, y (- 1) = 3, y^{^{'}} (- 1) = 6

y^{^{''}} + 4 y^{^{'}} + 4 y = 0, y (- 1) = 3, y^{^{'}} (- 1) = 5

y^{^{''}} + y^{^{'}} + 11 y = 0, y^{^{'}} (0) = 1, y (0) = 4

y^{^{''}} + 2 y^{^{'}} - 8 y = 0, y (0) = 3

What is the solution for y^{''}-12y^'+45y=0,y(0)=5,y^'(0)=-3 ?
The solution for y^{''}-12y^'+45y=0,y(0)=5,y^'(0)=-3 is y=e^{6t}(5cos(3t)-11sin(3t))