66.1.34 problem Problem 48

Internal problem ID [13881]
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
Date solved : Tuesday, January 28, 2025 at 06:07:08 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} y&=x y^{\prime }+{y^{\prime }}^{2} \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=-1 \end{align*}

✓ Solution by Maple

Time used: 0.879 (sec). Leaf size: 61

dsolve([y(x)=x*diff(y(x),x)+diff(y(x),x)^2,y(1) = -1],y(x), singsol=all)
 

\begin{align*} y &= -\frac {1}{2}+\frac {i \left (x -1\right ) \sqrt {3}}{2}-\frac {x}{2} \\ y &= \frac {\left (1+i \sqrt {3}\right ) \left (i \sqrt {3}-2 x +1\right )}{4} \\ y &= \frac {\left (i \sqrt {3}-1\right ) \left (i \sqrt {3}+2 x -1\right )}{4} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.007 (sec). Leaf size: 38

DSolve[{y[x]==x*D[y[x],x]+D[y[x],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*}