34.2.9 problem 9

Internal problem ID [6039]
Book : A treatise on ordinary and partial differential equations by William Woolsey Johnson. 1913
Section : Chapter 2, Equations of the first order and degree. page 20
Problem number : 9
Date solved : Wednesday, March 05, 2025 at 12:10:26 AM
CAS classification : [_separable]

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

✓ Maple. Time used: 0.008 (sec). Leaf size: 44

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

\[ y = x^{-\frac {2}{a}} {\mathrm e}^{\frac {-a \operatorname {LambertW}\left (-\frac {x^{-\frac {2}{a}} {\mathrm e}^{-\frac {2 c_1}{a}}}{a}\right )-2 c_1}{a}} \]

✓ Mathematica. Time used: 60.019 (sec). Leaf size: 29

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

\[ y(x)\to -a W\left (-\frac {e^{\frac {c_1}{a}} x^{-2/a}}{a}\right ) \]

✓ Sympy. Time used: 0.419 (sec). Leaf size: 20

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

\[ y{\left (x \right )} = - a W\left (- \frac {e^{\frac {C_{1} - 2 \log {\left (x \right )}}{a}}}{a}\right ) \]