Question

How do you find all solutions of the differential equation `dy/dx=xy+x+y+1`?

Answer 1

I gor: `y=e^(x^2/2+x+c)-1`

Explanation:

I would try to rearrange it as:
`(dy)/(dx)=y(x+1)+(x+1)`
and:
`(dy)/(dx)=(x+1)(y+1)`
I will then separate it to get:
`(dy)/(y+1)=(x+1)dx`
and integrate both sides:
`int(dy)/(y+1)=int(x+1)dx`
solve the integrals:
`ln(y+1)=x^2/2+x+c`
rearrange (taking the exponential of both sides to get rid of the `ln`):
`y+1=e^(x^2/2+x+c)`
and:
`y=e^(x^2/2+x+c)-1`