# GATE Papers >> Mechanical >> 2014 >> Question No 237

Question No. 237

Consider two solutions $x\left(t\right)={x}_{1}\left(t\right)$ and $x\left(t\right)={x}_{2}\left(t\right)$ of the differential equation

$\frac{{d}^{2}x\left(t\right)}{d{t}^{2}}+x\left(t\right)=0,t>0,$such that ${x}_{1}\left(0\right)=1,{\overline{)\frac{d{x}_{1}\left(t\right)}{dt}}}_{t=0}=0,{x}_{2}\left(0\right)=0,{\overline{)\frac{d{x}_{2}\left(t\right)}{dt}}}_{t=0}=1.$

The Wronskian $W\left(t\right)=\left|\begin{array}{cc}{x}_{1}\left(t\right)& {x}_{2}\left(t\right)\\ \frac{d{x}_{1}\left(t\right)}{dt}& \frac{d{x}_{2}\left(t\right)}{dt}\end{array}\right|\phantom{\rule{0ex}{0ex}}\phantom{\rule{0ex}{0ex}}$ at is