A unique solution to the differential equation $y = x \frac{dy}{dx} – (\frac{dy}{dx})^2$ passing through $(x_0,y_0)$ doesnot exist
then choose the correct option
$1.$ if $ x_0^2 > 4y_0$
$2.$ if $ x_0^2 = 4y_0$
$3.$ if $ x_0^2 < 4y_0$
$4.$ for any $(x_0 , y_0)$
My attempt : Here $y = x \frac{dy}{dx} – (\frac{dy}{dx})^2$
Now i put $x= e^z$ then $z= \log x$
So $y= Dy- D^2y$
$D^2y-Dy -y=0$
so $(D^2-D-1)y=0$
so auxiliary equation will be $m^2-m-1=0$ ,$m= \frac{1 +_{-}\sqrt – 3}{2}$
so $y= e^{\frac{1}{2}x} (c_1 \cos(\frac{\sqrt – 3}{2} ) + c_2\sin ({\frac{\sqrt – 3}{2}} )x)$
After that im not able to proceed further
Best Answer
Note that $y=bx-b^2$ is a solution, where $$ b=\frac{-x_0\pm\sqrt{x_0^2-4y_0}}{2} $$ This gives us two distinct solutions whenever $x_0^2\neq 4y_0$. If instead $x_0^2=4y_0$, we can still find a second solution $y=x^2/4$.