69.1.65 problem 92

Internal problem ID [14147]
Book : DIFFERENTIAL and INTEGRAL CALCULUS. VOL I. by N. PISKUNOV. MIR PUBLISHERS, Moscow 1969.
Section : Chapter 8. Differential equations. Exercises page 595
Problem number : 92
Date solved : Monday, March 31, 2025 at 12:10:44 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.112 (sec). Leaf size: 69

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

\begin{align*} y &= -i x \\ y &= i x \\ y &= 0 \\ y &= \sqrt {c_1 \left (c_1 -2 x \right )} \\ y &= \sqrt {c_1 \left (c_1 +2 x \right )} \\ y &= -\sqrt {c_1 \left (c_1 -2 x \right )} \\ y &= -\sqrt {c_1 \left (c_1 +2 x \right )} \\ \end{align*}

✓ Mathematica. Time used: 0.484 (sec). Leaf size: 126

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

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

✗ Sympy

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

Timed Out