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?