Question

Solution of the equation $y = x\frac{dy}{dx} + \frac{dx}{dy}$ represents

• A. Family of straight lines and a parabola
• B. Family of straight lines and a hyperbola
• C. family of circles and parabola
• D. None

Answer 1

Answer: (A)

Family of straight lines and a parabola

Answer 2

Putting $\frac{dy}{dx} = P$ we get

$y = xP + \frac{1}{P}$

Differentiation w.r.t. to x we get

$\frac{dy}{dx} = P + x \cdot \frac{dP}{dx} - \frac{1}{P^{2}}\frac{dP}{dx}$

̃ $P = P + x \cdot \frac{dP}{dx} - \frac{1}{P^{2}}\frac{dP}{dx}$ $\left\{ \because\frac{dy}{dx} = P \right\}$

̃ $\frac{dP}{dx} = 0$ or $P^{2} = \frac{1}{x}$

̃ P = C or $\left( \frac{dy}{dx} \right)^{2} = \frac{1}{x}$

Put these value in given equation we get $y = Cx + \frac{1}{C}$ which is equation of family of straight lines &

$y^{2} = \left( Px + \frac{1}{P} \right)^{2} = P^{2}x^{2} + 2x + \frac{1}{P^{2}}$

Put value of P² we get

y² = x + 2x + x = 4x

which is a singular solution and it represents parabola