# separable equation 2

• Mar 26th 2010, 05:57 PM
nameck
separable equation 2
By letting v = y/x, solve the following differential equation below by reducing them to equations where the variables are separable.

the solutions as the attachment..
the problem is, how to convert from..
dy/dx = (x^2 + y^2) / 2x^2

to...
xdv/dx = (v-1)^2 / 2
???
• Mar 26th 2010, 06:10 PM
dwsmith
Instead start by saying y=vx; therefore, dy=xdv+vdx.

Now take your de and multiple by dx to both sides and you will be left with dy = all that stuff.

Subtract all that stuff to obtain:

dy-(blah)dx=0

From taking the derivative of y=vx, we know dy=xdv+vdx. Substitute that in where you see dy and sub in vx where ever there are xs.
• Mar 26th 2010, 09:57 PM
nameck
still can't understand...(Worried)
• Mar 26th 2010, 10:08 PM
dwsmith
I typed up some of the steps in maple and converted to pdf.

You should be able to finish it from there.
• Mar 26th 2010, 10:17 PM
11rdc11
Quote:

Originally Posted by nameck
By letting v = y/x, solve the following differential equation below by reducing them to equations where the variables are separable.

the solutions as the attachment..
the problem is, how to convert from..
dy/dx = (x^2 + y^2) / 2x^2

to...
xdv/dx = (v-1)^2 / 2
???

$\displaystyle \frac{dy}{dx} = \frac{x^2+y^2}{2x^2} = \frac{x^2}{2x^2} +\frac{y^2}{2x^2}= \frac{1+v^2}{2}$

now since when sub in v we have to find

$\displaystyle v = \frac{y}{x}$

$\displaystyle vx =y$

$\displaystyle \frac{dv}{dx}x + v = \frac{dy}{dx}$

now put it all together

$\displaystyle \frac{dv}{dx}x + v = \frac{1+v^2}{2}$

$\displaystyle \frac{dv}{dx}x = \frac{1+v^2}{2} - v$

$\displaystyle \frac{dv}{dx}x = \frac{1+v^2}{2} - \frac{2v}{2}$

$\displaystyle \frac{dv}{dx}x = \frac{v^2-2v +1}{2}$

$\displaystyle \frac{dv}{dx}x = \frac{(v-1)^2}{2}$

$\displaystyle \int\frac{dv}{(v-1)^2} = \int\frac{dx}{2x}$
• Mar 26th 2010, 10:17 PM
mr fantastic
Quote:

Originally Posted by nameck
By letting v = y/x, solve the following differential equation below by reducing them to equations where the variables are separable.

the solutions as the attachment..
the problem is, how to convert from..
dy/dx = (x^2 + y^2) / 2x^2

to...
xdv/dx = (v-1)^2 / 2
???

$\displaystyle y = xv \Rightarrow \frac{dy}{dx} = v + x \frac{dv}{dx}$ using the product rule.

And obviously $\displaystyle y = xv \Rightarrow \frac{x^2 + y^2}{2x^2} = \frac{x^2 + x^2 v^2}{2x^2} = ....$
• Mar 26th 2010, 10:23 PM
Prove It
If $\displaystyle v = \frac{y}{x}$ then $\displaystyle y = vx$.

If you take the derivative of both sides, you get

$\displaystyle \frac{d}{dx}(y) = \frac{d}{dx}(vx)$

$\displaystyle \frac{dy}{dx} = v\,\frac{d}{dx}(x) + x\,\frac{d}{dx}(v)$

$\displaystyle \frac{dy}{dx} = v + x\,\frac{dv}{dx}$.

Putting this information into your DE:

$\displaystyle \frac{dy}{dx} = \frac{x^2 + y^2}{2x^2}$

$\displaystyle v + x\,\frac{dv}{dx} = \frac{x^2 + \left(vx\right)^2}{2x^2}$

$\displaystyle v + x\,\frac{dv}{dx} = \frac{x^2 + v^2x^2}{2x^2}$

$\displaystyle v + x\,\frac{dv}{dx} = \frac{1 + v^2}{2}$

$\displaystyle x\,\frac{dv}{dx} = \frac{1 + v^2}{2} - v$

$\displaystyle x\,\frac{dv}{dx} = \frac{v^2 - 2v + 1}{2}$

$\displaystyle x\,\frac{dv}{dx} = \frac{(v- 1)^2}{2}$

$\displaystyle \frac{2}{(v - 1)^2}\,\frac{dv}{dx} = \frac{1}{x}$

$\displaystyle \int{\frac{2}{(v - 1)^2}\,\frac{dv}{dx}\,dx} = \int{\frac{1}{x}\,dx}$

$\displaystyle \int{2(v - 1)^{-2}\,dv} = \int{\frac{1}{x}\,dx}$.

You should be able to go from here.