JEE 2022 — Mathematics PYQ

1. Identify the type of Differential Equation

The given equation is:

Rearranging for $\frac{dy}{dx}$:

This is a homogeneous differential equation because the degree of each term is the same.

2. Substitution

Let $y=vx$. Then, differentiating with respect to $x$:

Substitute these into the equation:

3. Variable Separation

Subtract $v$ from both sides:

4. Integration

Integrate both sides:

Using the standard formula $\int \frac{dx}{\sqrt{x^2+a^2}}=\ln |x+\sqrt{x^2+a^2}|$:

Substitute back $v=\frac{y}{x}$:

5. Apply Initial Condition $y(1)=3$

Substitute $x=1$ and $y=3$:

So, the particular solution is:

6. Find $y(2)$

Substitute $x=2$:

Squaring both sides:

Conclusion:

The value of $y(2)$ is 15.

The correct option is (A).