What is the solution for y^{''}+81y=0,y(0)=1,y^'(0)=7 ?

The solution for y^{''}+81y=0,y(0)=1,y^'(0)=7 is y=cos(9t)+7/9 sin(9t)

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y^{''}+81y=0,y(0)=1,y^'(0)=7

y^{^{''}} + 81 y = 0, y (0) = 1, y^{^{'}} (0) = 7

y = cos (9 t) + \frac{7}{9} sin (9 t)

Solution steps

y^{''} (t) + 81 y = 0

Solve linear ODE:

y = cos (9 t) + \frac{7}{9} sin (9 t)

y = cos (9 t) + \frac{7}{9} sin (9 t)

2 y^{^{''}} + 4 y^{^{'}} + 6 y = 0

\frac{d ^{2} y}{d x ^{2}} - 4 \frac{d}{d x} (4) + 4 y = 0

- 0.0375 x^{^{''}} + 0.4535 x^{^{'}} + 3.34 x = 0

y^{^{'}} = 7 y^{^{''}}

6 y^{^{'''}} + 78 y^{^{'}} - 696 y = 0

What is the solution for y^{''}+81y=0,y(0)=1,y^'(0)=7 ?
The solution for y^{''}+81y=0,y(0)=1,y^'(0)=7 is y=cos(9t)+7/9 sin(9t)