Question

$The solution of \frac{d y}{d x} = \frac{x}{y} is-$

$\frac{y^{2}}{2} - \frac{x^{2}}{2} = k$
$\frac{x^{2}}{2} + \frac{y^{2}}{2} = k$
$\frac{x - y}{2} = k$
$\frac{x + y}{5} + k$

Answer: $\frac{y^{2}}{2} - \frac{x^{2}}{2} = k$

Solution :-

Given that \frac{d y}{d x} = \frac{x}{y}

Now, use separation of variable method, we get

y \cdot d y = x \cdot d x \dots \dots (1)

Integrating both sides of eq(1), we get

\begin{aligned} \int y \cdot d y & = & \int x \cdot d x \\ \frac{y^{1 + 1}}{1 + 1} & = & \frac{x^{1 + 1}}{1 + 1} + k \\ \frac{y^{2}}{2} & = & \frac{x^{2}}{2} + k \\ \frac{y^{2}}{2} - \frac{x^{2}}{2} & = & k \cdot \end{aligned}

So, correct answer is option-1, $\to \frac{y^{2}}{2} - \frac{x^{2}}{2} = k$

#Ordinary Differential Equation #BSEB PYQ 2017 #Mathematics #BSEB PYQs