4.80. Integrate the Clairaut's equation

$$
y=x \frac{d y}{d x}+\frac{1}{2(d y / d x)}
$$
Solution. Let $p=\frac{d y}{d x}$, then from the equation we get $y=x p+\frac{1}{2 p}$. Differentiate both sides of the last equation and replace $d y$ with $p d x: p d x=p d x+x d p-\frac{d p}{2 p^{2}}$. Transform the obtained equation: $d p\left(x-\frac{1}{2 p^{2}}\right)=0$. From this it follows: a) $d p=0, p=C=$ const and the general solution of the Clairaut equation has the form $y=C x+\frac{1}{2 C}$; b) from the equality of the second factor to zero we get $x=\frac{1}{2 p^{2}}$; eliminate the parameter $p$ from this equation and the equation $y=x p+\frac{1}{2 p}$, we have $y^{2}=2 x$.

## 4.2 .5 .2 .

PROBLEMS FOR INDEPENDENT SOLUTION WITH ANSWERS

## Integrate the Lagrange and Clairaut equations.