22.1.82 problem 85

Internal problem ID [4388]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 2. First-Order and Simple Higher-Order Differential Equations. Page 78
Problem number : 85
Date solved : Tuesday, September 30, 2025 at 07:27:07 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _Clairaut]

\begin{align*} 2 {y^{\prime }}^{2} \left (y-x y^{\prime }\right )&=1 \end{align*}

✓ Maple. Time used: 0.044 (sec). Leaf size: 51

ode:=2*diff(y(x),x)^2*(y(x)-x*diff(y(x),x)) = 1; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} y &= \frac {3 x^{{2}/{3}}}{2} \\ y &= -\frac {3 x^{{2}/{3}} \left (1+i \sqrt {3}\right )}{4} \\ y &= \frac {3 x^{{2}/{3}} \left (-1+i \sqrt {3}\right )}{4} \\ y &= c_1 x +\frac {1}{2 c_1^{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.008 (sec). Leaf size: 67

ode=2*(D[y[x],x])^2*(y[x]-x*D[y[x],x])==1; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to c_1 x+\frac {1}{2 c_1{}^2}\\ y(x)&\to \frac {3 x^{2/3}}{2}\\ y(x)&\to -\frac {3}{2} \sqrt [3]{-1} x^{2/3}\\ y(x)&\to \frac {3}{2} (-1)^{2/3} x^{2/3} \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((-2*x*Derivative(y(x), x) + 2*y(x))*Derivative(y(x), x)**2 - 1,0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

Timed Out