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

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

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(d^2f)/(dx^2)=-a^2f(x)

\frac{d ^{2} f}{d x ^{2}} = - a^{2} f (x)

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

Solution steps

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

Solve linear ODE:

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

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

\frac{d ^{2} y}{d x ^{2}} - 36 y = 0

2 y^{^{''}} + y^{^{'}} + 3 y = 0

(D^{6} + 6 D^{4} + 9 D^{2}) y = 0

y^{^{''}} - 2 y^{^{'}} - 3 y = 0, y (0) = 5

x^{^{'''}} - x^{^{''}} = 0, x (0) = 1, x^{^{'}} (0) = 3, x^{^{''}} (0) = 2

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