Science:Math Exam Resources/Courses/MATH101/April 2017/Question 01 (d)/Solution 1

Since the differential equation ${\frac {dy}{dx}}={\frac {\cos x}{2y}}$ is separable, we can rewrite it as $ydy={\frac {\cos xdx}{2}}$.

We integrate both sides

$\int y{dy}=\int {\frac {\cos xdx}{2}}$

and thus

${\frac {y^{2}}{2}}={\frac {\sin x}{2}}+C$.

This yields

$y(x)=\pm {\sqrt {\sin x+2C}}$.

Since $y(0)={\sqrt {3}}>0$ , this means that we must take the positive root and that $2C=3$. Evaluate at $x={\frac {\pi }{2}}$ and obtain

Answer: $\color {blue}y\left({\frac {\pi }{2}}\right)=2$