69.1.63 problem 90

Internal problem ID [14216]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 90
Date solved : Tuesday, January 28, 2025 at 06:22:21 AM
CAS classification : [[_homogeneous, `class C`], _rational, _dAlembert]

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

✓ Solution by Maple

Time used: 0.102 (sec). Leaf size: 53

dsolve(y(x)=x*diff(y(x),x)^2+diff(y(x),x)^2,y(x), singsol=all)
 

\begin{align*} y &= 0 \\ y &= \frac {\left (x +1+\sqrt {\left (x +1\right ) \left (c_{1} +1\right )}\right )^{2}}{x +1} \\ y &= \frac {\left (-x -1+\sqrt {\left (x +1\right ) \left (c_{1} +1\right )}\right )^{2}}{x +1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.063 (sec). Leaf size: 57

DSolve[y[x]==x*(D[y[x],x])^2+(D[y[x],x])^2,y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to x-c_1 \sqrt {x+1}+1+\frac {c_1{}^2}{4} \\ y(x)\to x+c_1 \sqrt {x+1}+1+\frac {c_1{}^2}{4} \\ y(x)\to 0 \\ \end{align*}