23.2.69 problem 71

Internal problem ID [5424]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 2. THE DIFFERENTIAL EQUATION IS OF FIRST ORDER AND OF SECOND OR HIGHER DEGREE, page 278
Problem number : 71
Date solved : Tuesday, September 30, 2025 at 12:41:12 PM
CAS classification : [_dAlembert]

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

✓ Maple. Time used: 0.035 (sec). Leaf size: 183

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

\begin{align*} x +\frac {\left (-y a +\sqrt {a \left (a y^{2}+4 x \right )}\right ) \left (c_1 a +\operatorname {arcsinh}\left (\frac {y a}{2}-\frac {\sqrt {a \left (a y^{2}+4 x \right )}}{2}\right )\right )}{\sqrt {2 a^{2} y^{2}-2 a y \sqrt {a \left (a y^{2}+4 x \right )}+4 a x +4}\, a} &= 0 \\ x -\frac {\left (y a +\sqrt {a \left (a y^{2}+4 x \right )}\right ) \left (c_1 a +\operatorname {arcsinh}\left (\frac {y a}{2}+\frac {\sqrt {a \left (a y^{2}+4 x \right )}}{2}\right )\right )}{\sqrt {2 a^{2} y^{2}+2 a y \sqrt {a \left (a y^{2}+4 x \right )}+4 a x +4}\, a} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.312 (sec). Leaf size: 121

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

\[ \text {Solve}\left [\left \{x=\frac {\exp \left (\int _1^{K[1]}\frac {1}{K[2]^2 \left (K[2]+\frac {1}{K[2]}\right )}dK[2]\right ) \int \frac {\exp \left (-\int _1^{K[1]}\frac {1}{K[2]^2 \left (K[2]+\frac {1}{K[2]}\right )}dK[2]\right )}{K[1]+\frac {1}{K[1]}} \, dK[1]}{a}+c_1 \exp \left (\int _1^{K[1]}\frac {1}{K[2]^2 \left (K[2]+\frac {1}{K[2]}\right )}dK[2]\right ),y(x)=\frac {K[1]}{a}-\frac {x}{K[1]}\right \},\{y(x),K[1]\}\right ] \]

✗ Sympy

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

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