# first order non-linear differential equation

• Feb 2nd 2011, 03:12 PM
Empty
first order non-linear differential equation
I am trying to solve this differential equation and I can't get started and was wondering if I could get some help with a kick start:

$(\frac{dy}{dx})^2 +(\frac{dy}{dx})*x-\frac{dy}{dx}-x=0$

Any help would be greatly appreciated!

MT
• Feb 2nd 2011, 03:23 PM
Ackbeet
I know it sounds crazy, but couldn't you use the quadratic formula here to solve for y'? That is, you have

$(y')^{2}+y'(x-1)-x=0,$ so

$y'=\dfrac{-(x-1)\pm\sqrt{(x-1)^{2}-4(-x)}}{2}\dots$

Does that work?
• Feb 2nd 2011, 03:23 PM
TheEmptySet
Quote:

Originally Posted by Empty
I am trying to solve this differential equation and I can't get started and was wondering if I could get some help with a kick start:

$(\frac{dy}{dx})^2 +(\frac{dy}{dx})*x-\frac{dy}{dx}-x=0$

Any help would be greatly appreciated!

MT

Notice that

$\displaystyle \left(\frac{dy}{dx}\right)^2 +\left(\frac{dy}{dx}\right)x-\frac{dy}{dx}-x=0 \iff \left(\frac{dy}{dx}\right)^2 -\frac{dy}{dx}+\left(\frac{dy}{dx}\right)x-x=0$

factoring gives

$\displaystyle \frac{dy}{dx}\left( \frac{dy}{dx}-1\right)+x\left( \frac{dy}{dx}-1\right)=0$

$\displaystyle \left( \frac{dy}{dx}-1\right)\left( \frac{dy}{dx}+x\right)=0$

Use the zero product principle. This will give two different solutions. Remember for nonlinear equations that the principle of superposition does not hold. So you cannot add the solutions to obtain a more general solution.
• Feb 2nd 2011, 03:50 PM
Ackbeet
$y'=\dfrac{-x+1\pm|x+1|}{2}.$