QJEEM25Ma4452: JEE Main 2025 Mathematics - Class 12 JEEM25Ma4452 Solution: Let a curve pass through the points and . If the curve…

$\begin{aligned}& \frac{d y}{d x}=\frac{2(3+y) \cdot e^{2 x}}{7+e^{2 x}} \\& \frac{d y}{d x}-\frac{2 y \cdot e^{2 x}}{7+e^{2 x}}=\frac{6 \cdot e^{2 x}}{7+e^{2 x}} \\& \text { I.F. }=e^{-\int \frac{2 e^{2 x}}{7+e^{2 x}}}=\frac{1}{7+e^{2 x}}\end{aligned}$

$\begin{aligned}& \therefore y \cdot \frac{1}{7+\mathrm{e}^{2 \mathrm{x}}}=\int \frac{6 \mathrm{e}^{2 \mathrm{x}}}{\left(7+3^{2 x}\right)^2} \mathrm{dx} \\& \quad \frac{\mathrm{y}}{7+\mathrm{e}^{2 \mathrm{x}}}=\frac{-3}{7+\mathrm{e}^{2 \mathrm{x}}}+C \\& (0,5) \Rightarrow \frac{5}{8}=\frac{-3}{8}+C \Rightarrow C=1 \\& \therefore y=-3+7+\mathrm{e}^{2 \mathrm{x}} \\& y=\mathrm{e}^{2 \mathrm{x}}+4 \\& \therefore \mathrm{k}=8\end{aligned}$