Mathematics - Differential Equation Question with Solution | TestHub

$\begin{array}{rl}& \frac{dy}{dx}=\frac{2(3+y)\cdot e^{2x}}{7+e^{2x}}\\ & \frac{dy}{dx}-\frac{2y{e}^{2x}}{7+e^{2x}}=\frac{6\cdot e^{2x}}{7+e^{2x}}\\ & \text{ I.F. }=e^{-\int \frac{2e^{2x}}{7+e^{2x}}dx}\\ & \Rightarrow e^{-\ln \left(7+e^{2x}\right)}\\ & =\frac{1}{7+e^{2x}}\\ & y\cdot \frac{1}{7+e^{2x}}=\int \frac{6e^{2x}}{{\left(7+e^{2x}\right)}^2}dx\\ & \frac{y}{7+e^{2x}}=\frac{-3}{7+e^{2x}+C}\\ & \therefore y(0)=5\\ & \Rightarrow \frac{5}{8}=\frac{-3}{8}+C\\ & \Rightarrow C=1\\ & \therefore y=-3+7+e^{2x}\\ & y=e^{2x}+4\\ & \therefore k=8\end{array}$