Q83 / 150 · MHT CET 2025 (5 May Shift 2) · Mathematics

The solution of $\log \left(\frac{\mathrm{dy}}{\mathrm{d} x}\right)=2 \mathrm{x}-5 \mathrm{y}, \mathrm{y}(0)=0$ is

  • A
    $2 e^{2 x}+5 e^{5 y}=6$
  • Correct answer
    B
    $5 e^{2 x}-2 e^{5 y}=3$
  • C
    $2 e^{2 x}-5 e^{5 y}=6$
  • D
    $5 e^{2 x}+2 e^{5 y}=3$

Solution

The differential equation $\log \left(\frac{dy}{dx}\right) = 2x - 5y$ is equivalent to $\frac{dy}{dx} = e^{2x - 5y} = e^{2x}e^{-5y}$.

Separating variables yields $e^{5y} dy = e^{2x} dx$. Integrating both sides gives $\frac{e^{5y}}{5} = \frac{e^{2x}}{2} + C$.

Applying the initial condition $y(0) = 0$ produces $\frac{1}{5} = \frac{1}{2} + C$, so $C = -\frac{3}{10}$.

Thus, $\frac{e^{5y}}{5} = \frac{e^{2x}}{2} - \frac{3}{10}$. Multiplying through by 10 and rearranging terms leads to $5e^{2x} - 2e^{5y} = 3$.

This matches option $\boxed{\text{B}}$.

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

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