What is the solution for y^{''}-14y^'+53y=0,y(0)=6,y^'(0)=-2 ?

The solution for y^{''}-14y^'+53y=0,y(0)=6,y^'(0)=-2 is y=e^{7t}(6cos(2t)-22sin(2t))

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y^{''}-14y^'+53y=0,y(0)=6,y^'(0)=-2

y^{^{''}} - 14 y^{^{'}} + 53 y = 0, y (0) = 6, y^{^{'}} (0) = - 2

y = e^{7 t} (6 cos (2 t) - 22 sin (2 t))

Solution steps

y^{''} (t) - 14 y^{'} (t) + 53 y = 0

Solve linear ODE:

y = e^{7 t} (6 cos (2 t) - 22 sin (2 t))

y = e^{7 t} (6 cos (2 t) - 22 sin (2 t))

2 x^{^{''}} - 2 x = 0

y^{^{''}} - 7 y^{^{'}} - 10 y = 0

y^{^{''}} - 16 y^{^{'}} + 63 y = 0

y^{^{'''}} + 2 y = 0

y^{^{''}} - 10 y + 26 y = 0

What is the solution for y^{''}-14y^'+53y=0,y(0)=6,y^'(0)=-2 ?
The solution for y^{''}-14y^'+53y=0,y(0)=6,y^'(0)=-2 is y=e^{7t}(6cos(2t)-22sin(2t))