Calculate differential equation

[Chromosom]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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

 

[pasmith]

Homework Statement

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

Homework Equations

The Attempt at a Solution

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

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.

 

[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 it good answer?

 

[pasmith]

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.

This is correct.

And of course second solution: $$y=\frac{x^2}{4}+C$$

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