Question

(e^2x+5)dy+ye^2xdx=0

Answer 1

$(e^{2x}+5)dy+ye^{2x}dx=0 \\ \frac{dy}{y}=-\frac{e^{2x}dx}{e^{2x}+5}\\ \int\frac{dy}{y}=-\int\frac{e^{2x}dx}{e^{2x}+5}\\ \ln | y|=-\frac{1}{2}\int\frac{d(e^{2x}+5)}{e^{2x}+5}$

 

$\ln |y| =\ln \frac{1}{\sqrt{e^{2x}+5}} + \ln|C| \\ y=\frac{C}{\sqrt{e^{2x}+5}}$