JEE 2023 — Mathematics PYQ

1. Rearrange the Differential Equation

The given equation is:

Dividing by $dx$ and rearranging for $\frac{dy}{dx}$:

2. Use Substitution

Let $v=y^2$. Then $\frac{dv}{dx}=2y\frac{dy}{dx}$.

Substituting these into the equation:

This is a Linear Differential Equation of the form $\frac{dv}{dx}+P(x)v=Q(x)$.

3. Find the Integrating Factor (I.F.)

4. Solve the Linear Equation

Multiply the equation by the I.F. and integrate:

5. Apply Initial Conditions

Given $y(2)=\sqrt{\ln 2}$. Substitute $x=2$ and $y^2=\ln 2$:

So, the particular solution is:

6. Compare with the Given Form

The problem states the solution is in the form $\alpha x=\exp (x^{\beta }{y}^{\gamma })$.

Taking the natural log of both sides:

From our derived solution:

By comparing $\ln (2x)=x{y}^2$ with $\ln (\alpha x)=x^{\beta }{y}^{\gamma }$, we get:

• $\alpha =2$

• $\beta =1$

• $\gamma =2$

7. Final Calculation

Correct Option: (1) 1