What is the solution for 7x^{''}+3x^{''}+7x=0,x(0)=-4,x^'(0)=0 ?

The solution for 7x^{''}+3x^{''}+7x=0,x(0)=-4,x^'(0)=0 is x=-4cos(sqrt(7/10)t)

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7x^{''}+3x^{''}+7x=0,x(0)=-4,x^'(0)=0

7 x^{^{''}} + 3 x^{^{''}} + 7 x = 0, x (0) = - 4, x^{^{'}} (0) = 0

x = - 4 cos (\frac{7}{10} t)

Solution steps

7 x^{''} (t) + 3 x^{''} (t) + 7 x = 0

Solve linear ODE:

x = - 4 cos (\frac{7}{10} t)

x = - 4 cos (\frac{7}{10} t)

y^{^{'''}} + y^{^{''}} - 56 y^{^{'}} = 0

18 y^{^{''}} + (45 - 27) y^{^{'}} - 57 y = 0

y^{^{''}} - 10 y + 24 y = 0

y^{^{''''}} - 20 y^{^{'''}} + 100 y^{^{''}} = 0

81 y^{^{''}} - 9 y = 0, y (0) = 0, y^{^{'}} (0) = 0

What is the solution for 7x^{''}+3x^{''}+7x=0,x(0)=-4,x^'(0)=0 ?
The solution for 7x^{''}+3x^{''}+7x=0,x(0)=-4,x^'(0)=0 is x=-4cos(sqrt(7/10)t)