23.2.104 problem 106

Internal problem ID [5459]
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 : 106
Date solved : Tuesday, September 30, 2025 at 12:43:42 PM
CAS classification : [[_homogeneous, `class G`], _rational, _dAlembert]

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

✓ Maple. Time used: 0.016 (sec). Leaf size: 177

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

\begin{align*} -\frac {8 x \left (-\frac {3 c_1 \left (y-\sqrt {-4 x a +y^{2}}\right ) \sqrt {\frac {-y+\sqrt {-4 x a +y^{2}}}{x}}}{8}+x a -\frac {3 y^{2}}{4}+\frac {3 y \sqrt {-4 x a +y^{2}}}{4}\right )}{3 \left (y-\sqrt {-4 x a +y^{2}}\right )^{2}} &= 0 \\ -\frac {8 \left (\frac {3 c_1 \left (y+\sqrt {-4 x a +y^{2}}\right ) \sqrt {\frac {-2 y-2 \sqrt {-4 x a +y^{2}}}{x}}}{4}+x a -\frac {3 y^{2}}{4}-\frac {3 y \sqrt {-4 x a +y^{2}}}{4}\right ) x}{3 \left (y+\sqrt {-4 x a +y^{2}}\right )^{2}} &= 0 \\ \end{align*}

✓ Mathematica. Time used: 60.175 (sec). Leaf size: 4845

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

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✗ Sympy

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

Timed Out