23.2.122 problem 124

Internal problem ID [5477]
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 : 124
Date solved : Tuesday, September 30, 2025 at 12:44:30 PM
CAS classification : [[_homogeneous, `class G`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.019 (sec). Leaf size: 339

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

\begin{align*} \frac {4 \left (y 2^{-\frac {1}{a -1}} \left (a -\frac {1}{2}\right )^{2} a \sqrt {a^{2} y^{2}-4 x b}-\frac {b x 2^{\frac {a -2}{a -1}}}{4}+\left (\left (a -\frac {1}{2}\right )^{2} a y^{2}-2 x b \left (a -1\right )\right ) 2^{-\frac {1}{a -1}} a \right ) c_1 {\left (\frac {y a +\sqrt {a^{2} y^{2}-4 x b}}{x}\right )}^{\frac {1}{a -1}}+4 x \left (y \left (a -\frac {1}{2}\right ) \sqrt {a^{2} y^{2}-4 x b}+\left (a^{2}-\frac {1}{2} a \right ) y^{2}-2 x b \right ) a}{\left (2 a -1\right ) \left (y a +\sqrt {a^{2} y^{2}-4 x b}\right )^{2}} &= 0 \\ \frac {-4 \left (a^{2} y^{2}-a y \sqrt {a^{2} y^{2}-4 x b}-2 x b \right ) c_1 \left (a -\frac {1}{2}\right )^{2} {\left (-\frac {-y a +\sqrt {a^{2} y^{2}-4 x b}}{2 x}\right )}^{\frac {1}{a -1}}+4 x \left (-y \left (a -\frac {1}{2}\right ) \sqrt {a^{2} y^{2}-4 x b}+\left (a^{2}-\frac {1}{2} a \right ) y^{2}-2 x b \right ) a}{\left (2 a -1\right ) \left (y a -\sqrt {a^{2} y^{2}-4 x b}\right )^{2}} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 0.499 (sec). Leaf size: 143

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

\begin{align*} \text {Solve}\left [\frac {2 \left ((a-1) \log \left (\sqrt {a^2 y(x)^2-4 b x}+(a-1) y(x)\right )+a \log \left (\sqrt {a^2 y(x)^2-4 b x}-a y(x)\right )\right )}{2 a-1}=c_1,y(x)\right ]\\ \text {Solve}\left [\frac {2 \left ((a-1) \log \left (\sqrt {a^2 y(x)^2-4 b x}-a y(x)+y(x)\right )+a \log \left (\sqrt {a^2 y(x)^2-4 b x}+a y(x)\right )\right )}{2 a-1}=c_1,y(x)\right ] \end{align*}

✗ Sympy

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

Timed Out