Answer:
[tex]y=C(x+3)^2[/tex]
Step-by-step explanation:
We are given:
[tex]\displaystyle (x+3)y^\prime=2y[/tex]
Separation of Variables:
[tex]\displaystyle \frac{1}{y}\frac{dy}{dx}=\frac{2}{x+3}[/tex]
So:
[tex]\displaystyle \frac{dy}{y}=\frac{2}{x+3} \, dx[/tex]
Integrate:
[tex]\displaystyle \int\frac{dy}{y}=\int\frac{2}{x+3}\, dx[/tex]
Integrate:
[tex]\displaystyle \ln|y|=2\ln|x+3|+C[/tex]
Raise both sides to e:
[tex]|y|=e^{2\ln|x+3|+C}[/tex]
Simplify:
[tex]|y|=(e^{\ln|x+3|})^2\cdot e^C[/tex]
So:
[tex]|y|=C|x+3|^2[/tex]
Simplify:
[tex]y=\pm C(x+3)^2=C(x+3)^2[/tex]
