solving the differential equation : $y' = \sqrt{4x+2y+1}$

Question

I tried solving the differential equation :

$$y' = \sqrt{4x+2y+1}$$

but I really have no direction as to how to solve it. The fact both $x$ and $y$ are under the same root makes for difficulty when solving, since there seems to be no way to separate the two variables. Ideas?

Answer 1

$$y' = \sqrt{4x+2y+1}$$ substitute $z=4x+2y+1$ $$z=4x+2y+1 \implies z'=4+2y' \implies y'=(z'-4)/2$$ $$z'-4=2\sqrt z \implies \int \frac {dz}{\sqrt z+2}=2x+K$$ Then substitute again $u=\sqrt z$ and solve: $$\int \frac {2udu}{u+1}=2x+K$$

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Or substitute $z^2=4x+2y+1 \implies zz'=2+y'$ And solve the seperable equation $$zz'-2=z \implies \int \frac {zdz}{z+2}=x+K$$

Answer 2

Substitute $$u(x)=4x+2y(x)$$ then you will get $$\frac{du(x)}{dx}=2\left(\sqrt{u(x)+1}+2\right)$$