Question

Find the general solution of (x+3)y’=2y

Answer 1

`y=C(x+3)^2`

Explanation:

We are given:

`\displaystyle (x+3)y^\prime=2y`

Separation of Variables:

`\displaystyle (1)/(y)(dy)/(dx)=(2)/(x+3)`

So:

`\displaystyle (dy)/(y)=(2)/(x+3) \, dx`

Integrate:

`\displaystyle \int(dy)/(y)=\int(2)/(x+3)\, dx`

Integrate:

`\displaystyle \ln|y|=2\ln|x+3|+C`

Raise both sides to e:

`|y|=e^(2\ln|x+3|+C)`

Simplify:

`|y|=(e^(\ln|x+3|))^2\cdot e^C`

So:

`|y|=C|x+3|^2`

Simplify:

`y=\pm C(x+3)^2=C(x+3)^2`