Question

What is the solution for the second order differential equation $\frac{d^{2}y}{d{x}^2}+y=0$, with the initial conditions $y{|}_{x=0}=5$ and ${\begin{array}{c}\frac{dy}{dx}\end{array}|}_{x=0}=10$?

• A. $y=5+0sinx$
• B. $y=5cos-5sinx$
• C. $y=5cosx+10x$
• D. $y=5cosx+10sinx$

Answer 1

The correct option is D $y=5cosx+10sinx$

Given that $\frac{d^{2}y}{d{x}^2}+y=0$ ... $\left(1\right)$
With $y\left(0\right)=5$ & $y^{\prime }\left(0\right)=10$
Auxiliary equation is
$m^2+1=0$
$m=0\pm i$
General solution $y=c_{1}cosx+c_{2}sinx$ ... $\left(2\right)$
Using $y\left(0\right)=5$ & $y^{\prime }\left(0\right)=10$, we get
$C_2=10$ & $C_1=5$
Hence by $\left(2\right)$,
$y=5cosx+10sinx$