Substitution Methods for first order differential equations

[smashyash]

I have the following equation::

xy' = y + 2*sqrt(xy)

I know I should either use the F(y/x) substitute or Bernoulli's method of substitution but I'm not sure how to manipulate the equation to determine which it is.

If someone had some helpful tips on how to start, please let me know!
Thanks!

 

[velvexor]

1. First, isolate the y' term by dividing by x.

2. Use the substitution y=zx and the fact that \frac{dy}{dx}=\frac{dz}{dx}x+z to get the original DE to separable form.

3. Solve and express your answer in terms of y and x.

 

[smashyash]

Thanks so much! Think I got it :)

 

[velvexor]

You're welcome! You can always check your answer by differentiating and substituting back to the original DE.

 

[smashyash]

thanks again! some of these are kind of tricky. I'm actually stuck on another now::

x(x+y)y' + y(3x+y) = 0

do you advise multiplying through these or not? I'm not very good at recognizing how to get started on these problems yet..

 

[smashyash]

Actually, I now figured out that if you divide x throughout the equation, you can get a nice bunch of (y/x) functions to substitute v with! My answer doesn't quite seem right though..

(1+v)y' + v(3+v) = 0

substitute v = (y/x) ; y = vx

y' = (v+xv')

(1+v)(v+xv') + v(3+v) = 0

v+xv' = (-3v-v^2)/(1+v)

xv' = [-2v(v+2)]/(1+v)

(v+1)/(-2v^2-4v) dv = dx/x

then integrate... but it's kind of nasty which leads me to believe it's not quite right.

 

[velvexor]

I think you need to make this equation exact by finding an integrating factor. Take a look at https://www.physicsforums.com/showthread.php?t=482259" to follow the steps, if you need to. I found the solution to be x^3y+\frac{x^2y^2}{2}=c

Also, methinks this is not the right section to post questions of this kind (i.e. HW questions). Next time, post similar questions under howework section.