Question

If $\frac{d^{2}y}{d{t}^2}+y=0$ under the conditions $y=1,\frac{dy}{dt}=0$, when t=0 then y is equal to

• A. $sint$
• B. $cost$
• C. $tant$
• D. $cott$

Answer 1

The correct option is B $cost$

$\frac{d^{2}y}{d{t}^2}+y=0\Rightarrow (D^2+1)y=0$ ...(i)
Wherer $D=\frac{d}{dt}$
$AE$ is $m^2+1=0\Rightarrow m=\pm i$
So $CF=e^{ot}(C_{1}cost+C_{2}sint]$ and $PI=0$
Hence General solutin is $y=CF+PI$
$y=C_{1}cost+C_{2}sint$ .... (ii)
$\frac{dy}{dx}=-C_{1}sint+C_{2}cost$ ....(iii)
Using $y\left(0\right)=1\Rightarrow 1=C_1+0\Rightarrow C_1=1$
Using $y^{\prime }\left(0\right)=0\Rightarrow 0=0+C_2\Rightarrow C_2=0$
So $y=cost$