What is the solution for (d^2y)/(dx^2)=-9y ?

The solution for (d^2y)/(dx^2)=-9y is y=c_{1}cos(3x)+c_{2}sin(3x)

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(d^2y)/(dx^2)=-9y

\frac{d ^{2} y}{d x ^{2}} = - 9 y

y = c_{1} cos (3 x) + c_{2} sin (3 x)

Solution steps

\frac{d ^{2} y}{d x ^{2}} = - 9 y

Solve linear ODE:

y = c_{1} cos (3 x) + c_{2} sin (3 x)

y = c_{1} cos (3 x) + c_{2} sin (3 x)

y^{^{''}} = 7 y^{^{'}} - 12 y, y^{^{'}} (0) = 1

x^{^{''}} + 8 x^{^{'}} + 15 x = 0

y^{^{''}} + 16 y^{^{'}} = 0, y (0) = 2, y^{^{'}} (0) = - 2

y^{^{'''}} + 7 y^{^{''}} - 17 y^{^{'}} + 9 y = 0

y^{^{''}} + y = 0, y (0) = 2, y^{^{'}} (0) = 1

What is the solution for (d^2y)/(dx^2)=-9y ?
The solution for (d^2y)/(dx^2)=-9y is y=c_{1}cos(3x)+c_{2}sin(3x)