Answer to Question #223125 in Differential Equations for Giren

Question

Find the particular solution of each of the differential equation expressing y explicitly in terms of x.

e) dy/dx - 2xy = 0 y=3 when x = 0

f) xdy/dx + y ²= 0 (x>0) y=1/2 when x= 1

Answer 1

$1\\ \frac{dy}{dx}-2xy=0\\ \frac{dy}{dx}=2xy\\ \frac{dy}{y}=2xdx\\ \text{Integrate both sides}\\ \ln y=x^2+C\\ y=Ae^{x^2}\\ \text{When } x=0, y=3\\ 3=Ae^0\\ A=3\\ \text{The particular solution is }\\ y=3e^{x^2}.\\\\ 2.\\ x\frac{dy}{dx}+y^2=0\\ xdy=-y^2dx\\ -\frac{dy}{y^2}=\frac{dx}{x}\\ \text{Integrate both sides}\\ \frac{1}{y}=\ln x+ C\\ y=\frac{1}{\ln x+ C}\\ \text{When } x=1, y= \frac{1}{2}\\ \frac{1}{2}=\frac{1}{\ln x+ C}\\ \frac{1}{2}=\frac{1}{C}\\ C=2\\ \text{The particular solution is }\\ y=\frac{1}{\ln x+2}$