23.2.68 problem 70

Internal problem ID [5423]
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 : 70
Date solved : Tuesday, September 30, 2025 at 12:41:10 PM
CAS classification : [_dAlembert]

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

✓ Maple. Time used: 0.095 (sec). Leaf size: 398

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

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

✓ Mathematica. Time used: 0.347 (sec). Leaf size: 133

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]
 

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

✗ 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