40.5.12 problem 28

Internal problem ID [6677]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 9. Equations of first order and higher degree. Supplemetary problems. Page 65
Problem number : 28
Date solved : Sunday, March 30, 2025 at 11:17:52 AM
CAS classification : [[_1st_order, _with_linear_symmetries], _dAlembert]

\begin{align*} {y^{\prime }}^{2}-x y^{\prime }-y&=0 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 77

ode:=diff(y(x),x)^2-x*diff(y(x),x)-y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\begin{align*} \frac {c_1}{\sqrt {2 x -2 \sqrt {x^{2}+4 y}}}+\frac {2 x}{3}+\frac {\sqrt {x^{2}+4 y}}{3} &= 0 \\ \frac {c_1}{\sqrt {2 x +2 \sqrt {x^{2}+4 y}}}+\frac {2 x}{3}-\frac {\sqrt {x^{2}+4 y}}{3} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 60.111 (sec). Leaf size: 1003

ode=D[y[x],x]^2-x*D[y[x],x]-y[x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} \text {Solution too large to show}\end{align*}

✗ Sympy

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

NotImplementedError : The given ODE -x/2 - sqrt(x**2 + 4*y(x))/2 + Derivative(y(x), x) cannot be solved by the factorable group method