# Differential Equations Help

## Homework Statement

Solve The following Equations:
2(y+3)dx-xydy=0

(x2-xy+y2)dx - xydy=0 use following assumption y=vx

xy3+ex2dy=0

## The Attempt at a Solution

I am still a novice at diff eqs but here is what I got on the first one:
After seperating it I ended up with
(dx/x)=(ydy)/(2y+6) Then I get stuck with integrating the side with the Y

For the other two I believe they can not be separated and I am not sure what to do when this is the case

rock.freak667
Homework Helper
For the right side, you can rewrite it as

y/2(y+3) dy or ½(y+3-3)/(y+3), you can simply it even further i.e. polynomial division

vela
Staff Emeritus
Homework Helper
For the y-integral on the first one, you can do this:$$\frac{1}{2}\int \frac{y}{y+3}\,dy = \frac{1}{2}\int \frac{(y+3)-3}{y+3}\,dy = \frac{1}{2}\int \left[1-\frac{3}{y+3}\right]\,dy$$or you could use the substitution u=y+3.

On the second one, what did you get when you used the substitution y=vx?

For the y-integral on the first one, you can do this:$$\frac{1}{2}\int \frac{y}{y+3}\,dy = \frac{1}{2}\int \frac{(y+3)-3}{y+3}\,dy = \frac{1}{2}\int \left[1-\frac{3}{y+3}\right]\,dy$$or you could use the substitution u=y+3.

On the second one, what did you get when you used the substitution y=vx?
I have never used the substitution method I have no clue how to use that I looked it up earlier because someone told me that but I was unable to use the examples to work that one out

Mark44
Mentor
xy3+ex2dy=0
I don't think you wrote this correctly - there seems to be a dx missing.

vela
Staff Emeritus
Homework Helper
I have never used the substitution method I have no clue how to use that I looked it up earlier because someone told me that but I was unable to use the examples to work that one out
If you differentiate y=vx with respect to x, you'll get
$$\frac{dy}{dx} = v + x\frac{dv}{dx}$$
Multiplying through by dx, you end up with
$$dy = v \,dx + x\, dv$$
Use this and the original substitution to eliminate y from the original equation. You should find it separates then, allowing you to solve for v, from which you can then find y.

I don't think you wrote this correctly - there seems to be a dx missing.
ahh you are correct it is suppose to be a dx after the xy3

If you differentiate y=vx with respect to x, you'll get
$$\frac{dy}{dx} = v + x\frac{dv}{dx}$$
Multiplying through by dx, you end up with
$$dy = v \,dx + x\, dv$$
Use this and the original substitution to eliminate y from the original equation. You should find it separates then, allowing you to solve for v, from which you can then find y.
I tried what you said and plugged stuff back in and then I Tried separating things out and I cant seem to get it to separate out I am stuck at
X2(1-V-V2)dx=x2v2+(x3v)dv/dx)

vela
Staff Emeritus
Homework Helper
Please show your work. It's impossible to see what went wrong without seeing what you actually did.

HallsofIvy
Homework Helper
For (2) you are told to let y= vx and from that dy= vdx+ xdv. Replace y and dy in the equation with those. It will reduce to a separable equation.

vela
Staff Emeritus