CBSE 2023 · Region 5 · Set 1 · Q38 · 4 marks

An equation involving derivatives of the dependent variable with respect to the independent variables is called a differential equation. A differential equation of the form $\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{F}(x, \mathrm{y})$ is said to be homogeneous if $\displaystyle \mathrm{F}(x, \mathrm{y})$ is a homogeneous function of degree zero, whereas a function $\displaystyle \mathrm{F}(x, \mathrm{y})$ is a homogenous function of degree n if $\displaystyle \mathrm{F}(\lambda x, \lambda \mathrm{y})=\lambda^{\mathrm{n}} \mathrm{F}(x, \mathrm{y})$. To solve a homogeneous differential equation of the type $\displaystyle \frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{F}(x, \mathrm{y})=$ $\displaystyle \mathrm{g}\left(\frac{\mathrm{y}}{x}\right)$, we make the substitution $\displaystyle \mathrm{y}=\mathrm{v} x$ and then separate the variables. Based on the above, answer the following questions :

(I)

Show that $\displaystyle \left(x^{2}-\mathrm{y}^{2}\right) \mathrm{d} x+2 x \mathrm{y} \mathrm{dy}=0$ is a differential equation of the type $\displaystyle \frac{\mathrm{dy}}{\mathrm{d} x}=\mathrm{g}\left(\frac{\mathrm{y}}{x}\right)$.

(II)

Solve the above equation to find its general solution.

CBSE Class 12 Mathematics past-paper question from the 2023board exam, with the answer as CBSE’s own marking scheme gives it. Where our answers come from.