Question

General solution of the differential equation $y=x\frac{dy}{dx}+\frac{dx}{dy}$ represents:

• A. a straight line or a hyperbola
• B. a straight line or a parabola
• C. a parabola or a hyperbola
• D. circles

Answer 1

Answer: (B)

a straight line or a parabola

Substitute $p=\frac{dy}{dx}$, the given equation can be written as $y=px+1/p$. Differentiating w.r.t. $x$, we have
$p=\frac{dy}{dx}=p+x\frac{dp}{dx}-\frac{1}{p^2}\frac{dp}{dx}$
$\Rightarrow (x-\left(1/p^2\right))\frac{dp}{dx}=0\Rightarrow p^2=\frac{1}{x}$ or $\frac{dp}{dx}=0$
If $\frac{dp}{dx}=0$ then $p=$ constant $=c$ putting this value in given equation, we get $y=cx+1/c$ which represents a straight line. If $p^2=1/x$ then $y^2={(px+1/p)}^2=p^2{x}^2+1/p^2+2x=\frac{1}{x}{x}^2+x+2x=4x$, which represents a parabola.