Internal problem ID [12145]
Book: Differential equations and the calculus of variations by L. ElSGOLTS. MIR PUBLISHERS,
MOSCOW, Third printing 1977.
Section: Chapter 1, First-Order Differential Equations. Problems page 88
Problem number: Problem 48.
ODE order: 1.
ODE degree: 2.
CAS Maple gives this as type [[_1st_order, _with_linear_symmetries], _Clairaut]
\[ \boxed {y-y^{\prime } x -{y^{\prime }}^{2}=0} \] With initial conditions \begin {align*} [y \left (1\right ) = -1] \end {align*}
✓ Solution by Maple
Time used: 0.391 (sec). Leaf size: 66
dsolve([y(x)=x*diff(y(x),x)+diff(y(x),x)^2,y(1) = -1],y(x), singsol=all)
\begin{align*} y \left (x \right ) &= -\frac {1}{2}+\frac {i \left (-1+x \right ) \sqrt {3}}{2}-\frac {x}{2} \\ y \left (x \right ) &= \frac {\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-2 x +1\right )}{4} \\ y \left (x \right ) &= \frac {\left (i \sqrt {3}-1\right ) \left (i \sqrt {3}+2 x -1\right )}{4} \\ \end{align*}
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 38
DSolve[{y[x]==x*y'[x]+y'[x]^2,{y[1]==-1}},y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to (-1)^{2/3}-\sqrt [3]{-1} x \\ y(x)\to \sqrt [3]{-1} \left (\sqrt [3]{-1} x-1\right ) \\ \end{align*}