# Calculate differential equation

1. Feb 14, 2014

### Chromosom

1. The problem statement, all variables and given/known data

$$y=xy^\prime-\left(y^\prime\right)^2$$

2. Relevant equations

3. The attempt at a solution

Unfortunately, I do not have any good idea. I tried $$y=xt(x)$$, but the equation only became worse.

2. Feb 14, 2014

### pasmith

Differentiate both sides with respect to $x$. You do have to check whether all the solutions of the resulting 2nd order ODE are solutions of this ODE.

3. Feb 14, 2014

### Chromosom

Very clever :)

$$y^\prime=y^\prime+xy^{\prime\prime}-2y^\prime y^{\prime\prime}$$

$$xy^{\prime\prime}-2y^\prime y^{\prime\prime}=0$$

$$y^{\prime\prime}\left(x-2y^\prime\right)=0$$

Now we could expect that $$y=ax+b$$, but:

$$y^\prime=a$$

$$ax+b=ax-a^2$$

$$b+a^2=0$$

The last equation must be satisfied in order to be a solution. And of course second solution: $$y=\frac{x^2}{4}+C$$

Is there any restriction on the value of $C$?