Question

Solve 2x(y+2sqrt(x))y'=(y+sqrt(x))^2 explicitly.

Answer 1

\[\begin{align*}
2x(y+2\sqrt x)y'&=(y+\sqrt x)^2\\
-(y+\sqrt x)^2+2x(y+2\sqrt x)y'&=0
\end{align*}\]
The equation \(M(x,y)+N(x,y)y'=0\) is exact if \(\dfrac{\partial M}{\partial y}=\dfrac{\partial N}{\partial x}\).
\[\frac{\partial }{\partial y}\left[-(y+\sqrt x)^2\right]=-2(\sqrt x+y)\]
\[\frac{\partial }{\partial x}\left[2x(y+2\sqrt x)\right]=2(3\sqrt x+y)\]
Find an integrating factor of the form,
\[\mu(x)=\exp\left(\int\frac{\frac{\partial M}{\partial y}-\frac{\partial N}{\partial x}}{N}~dx\right)\]
if it exists...
\[\begin{align*}\mu(x)&=\exp\left(\int\frac{-2(\sqrt x+y)-2(3\sqrt x+y)}{2x(y+2\sqrt x)}~dx\right)\\\\
&=\exp\left(\int\frac{-4y-8\sqrt x}{2x(y+2\sqrt x)}~dx\right)\\\\
&=\exp\left(-2\int\frac{dx}{x}\right)\\\\
&=\exp(-2\ln x)\\\\
&=\frac{1}{x^2}\end{align*}\]
Here's the original equation with the IF
\[\begin{align*}
-\frac{(y+\sqrt x)^2}{x^2}+\frac{2(y+2\sqrt x)}{x}y'&=0\end{align*}\]
Verify that this new equation \(M^*(x,y)+N^*(x,y)y'=0\) is exact.
\[\frac{\partial }{\partial y}\left[-\frac{(y+\sqrt x)^2}{x^2}\right]=-\frac{2(y+\sqrt x)}{x^2}\]
\[\frac{\partial }{\partial x}\left[\frac{2(y+2\sqrt x)}{x}\right]=-\frac{2(y+\sqrt x)}{x^2}\]

Answer 2

Shouldn't this equation be solved using the substitution y=x^(1/2)*t?

Answer 3

I didn't mean to say the exact-equation approach was the ONLY approach. I just hadn't considered a substitution.

Answer 4

But suppose we try your substitution: \(y=t\sqrt x\), then \(y'=\sqrt x~t'+\dfrac{t}{2\sqrt x}\).
\[\begin{align*}
2x(y+2\sqrt x)y'&=(y+\sqrt x)^2\\\\
2x(t+2)\left(xt'+\frac{t}{2}\right)&=x(t+1)^2\\\\
t'&=\frac{(t+1)^2-t(t+2)}{2x(t+2)}\\\\
t'&=\frac{1}{2x(t+2)}\\\\
(t+2)~t'&=\frac{1}{2x}
\end{align*}\]
It works quite nicely!

Answer 5

Wait a minute. My computer doesn't really work. Please don't leave.

Answer 6

Just a heads up then, I have a class to get to in 30 minutes.

Answer 7

Okay.

Answer 8

When will you come back after your class ends?

Answer 9

Maybe in like 3 hours? I have work to do afterwards.

Answer 10

So you only have time in the morning, right?

Answer 11

Free time isn't exactly fixed for me :P

I come on the site whenever I can come on the site. It's 2:00 PM where I am right now.

Answer 12

The problem is that I can't see your response since Processing math: 0%. :P

Answer 13

Yeah there's always some problem with this page... Here's a snapshot.

Answer 14

Thank you so much for the help and snapshot. :)