69.1.63 problem 90

Internal problem ID [14137]
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 : Wednesday, March 05, 2025 at 10:36:31 PM
CAS classification : [[_homogeneous, `class C`], _rational, _dAlembert]

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

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

ode:=y(x) = x*diff(y(x),x)^2+diff(y(x),x)^2; 
dsolve(ode,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*}

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

ode=y[x]==x*(D[y[x],x])^2+(D[y[x],x])^2; 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 0.886 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*Derivative(y(x), x)**2 + y(x) - Derivative(y(x), x)**2,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {C_{1}^{2}}{4} - C_{1} \sqrt {x + 1} + x + 1 \]