Question

найти решение дифференциального уравнения​

Answer 1

$1)\; \; ye^{x}\, dx-(1+e^{x})\, dy=0\\\\\int \frac{e^{x}\, dx}{1+e^{x}}=\int \frac{dy}{y}\\\\ln(1+e^{x})=ln|y|+lnC\\\\Cy=1+e^{x}\\\\y=\frac{1}{C}\cdot (1+e^{x})\; \; \to \; \; \underline {y=C^*\cdot (1+e^{x})}\; ,\; C^*=\frac{1}{C}$

$2)\; \; y'=4x^{-3}\\\\\frac{dy}{dx}=\frac{4}{x^3}\\\\\int dy=\int \frac{4\, dx}{x^3}\\\\y=4\cdot \frac{x^{-2}}{-2}+C\\\\y=-\frac{2}{x^2}+C\\\\y(1)=2\, :\; \; 2=-\frac{2}{1^2}+C\; \; ,\; \; C=4\; ,\\\\\underline {y=-\frac{2}{x^2}+4}$