Question

Match each differential equation in Group I to its family of solution curves from Group II

Group I

A. $\frac{dy}{dx}=\frac{y}{x}$

B. $\frac{dy}{dx}=\frac{-y}{x}$

C. $\frac{dy}{dx}=\frac{x}{y}$

D. $\frac{dy}{dx}=\frac{-x}{y}$

Group II

1. Circles

2. Straight lines

3. Hyperbolas

• A. A-2, B-3, C-3, D-1
• B. A-1, B-3, C-2, D-1
• C. A-2, B-1, C-3, D-3
• D. A-3, B-2, C-1, D-2

Answer 1

The correct option is A A-2, B-3, C-3, D-1

$\frac{dy}{dx}=\frac{y}{x}$

$\frac{dy}{y}=\frac{dx}{x}$

Integrating both sides

log y = log x + log c

$\Rightarrow y=cx$

Which is equation of family of straight lines.

$\frac{dy}{dx}=-\frac{y}{x}$

$\frac{dy}{y}=-\frac{dx}{x}$

Integrating both sides

log y = -log x + log c

xy = c

Which is equation of family of hyperbolas.

$\frac{dy}{dx}=\frac{x}{y}$

ydy = x dx

integrating both sides.

$x^2-y^2=c$

which is equation of family of hyperbolas

$\frac{dy}{dx}=-\frac{x}{y}$

xdx + ydy = 0

Integrating

$x^2+y^2=c$

which is equation of family of circles.