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

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

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

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

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

Solution steps

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

Solve linear ODE:

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

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

y^{^{''}} + 4 y = 0, y (0) = 2, y^{^{'}} (0) = 2

49 y^{^{''}} - 126 y^{^{'}} + 117 y = 0, y (4) = 2, y^{^{'}} (4) = 2

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

2 y^{^{''}} - 10 y^{^{'}} + 12 y = 0

y^{^{'''}} - 3 y^{^{''}} - 49 y^{^{'}} - 45 y = 0

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