Q56 / 150 · MHT CET 2025 (21 Apr Shift 1) · Mathematics

The general solution of $\frac{\mathrm{dy}}{\mathrm{d} x}=2 x \mathrm{ye}^{x^2}$ is

  • A
    $y=e^{-e^{x^2}} c, \quad$ where $c$ is the constant of integration
  • B
    $y=e^{-x^2} c, \quad$ where $c$ is the constant of integration
  • Correct answer
    C
    $\mathrm{y}=\mathrm{e}^{\mathrm{e}^{x^2}} \mathrm{c}, \quad$ where c is the constant of integration
  • D
    $\mathrm{y}=\mathrm{e}^{x^2} \mathrm{c}, \quad$ where c is the constant of integration

Solution

The separable differential equation is $\frac{dy}{dx} = 2xye^{x^2}$.

Separating variables yields $\frac{dy}{y} = 2xe^{x^2}\,dx$.

Integration gives $\ln|y| + C_1 = \int e^u\,du$ where $u = x^2$ and $du = 2x\,dx$, resulting in $\ln|y| + C_1 = e^{x^2} + C_2$.

Combining constants and exponentiating:

$|y| = e^{e^{x^2} + C}$

$|y| = e^{e^{x^2}}e^C$

Absorbing constants gives $y = ce^{e^{x^2}}$, with $c \neq 0$ including $y = 0$ as a solution when $c = 0$.

This matches option C.

The answer is $\boxed{C}$.

Hence, the correct answer is option: 'C'.

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