23.2.118 problem 120

Internal problem ID [5473]
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 : 120
Date solved : Tuesday, September 30, 2025 at 12:44:27 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 33

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

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

✗ Mathematica

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

{}

✓ Sympy. Time used: 1.214 (sec). Leaf size: 82

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

\[ \left [ \log {\left (x \right )} = C_{1} + \begin {cases} - \operatorname {acosh}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {for}\: \left |{\frac {y^{2}{\left (x \right )}}{a x^{2}}}\right | > 1 \\i \operatorname {asin}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {otherwise} \end {cases}, \ \log {\left (x \right )} = C_{1} + \begin {cases} \operatorname {acosh}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {for}\: \left |{\frac {y^{2}{\left (x \right )}}{a x^{2}}}\right | > 1 \\- i \operatorname {asin}{\left (\frac {y{\left (x \right )}}{\sqrt {a} x} \right )} & \text {otherwise} \end {cases}\right ] \]