Question

Verify that the given function is a solution of the differential equation. y = c₁ sin x + c₂ cos x; $\frac{d^{2}y}{dx^{2}}+y=0$

Answer 1

To verify that the given function is a solution of the differential equation. y = c₁ sin x + c₂ cos x; $\frac{d^{2}y}{dx^{2}}+y=0$

We first calculate it's 1st order derivative,than second order derivative

$y = c_{1}\: sinx + c_{2} \: cosx ...eq1\\ \\ \frac{dy}{dx} = c_{1} \: cos \: x - c_{2} \: sin \: x \: ...eq2 \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} } = - c_{1} \: sin \: x - c_{2} \: cos \: x..eq3$
from eq 1 put the value in eq3

$\frac{ {d}^{2}y }{ {dx}^{2} } = - (c_{1} \: sin \: x + c_{2} \: cos \: x) \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} } = - \: y \\ \\ \frac{ {d}^{2}y }{ {dx}^{2} } + y \: = 0$

So,here we can see that this is the desired equation.

hence verified.