87.3.15 problem 17

Internal problem ID [23264]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 1. Elementary methods. First order differential equations. Exercise at page 26
Problem number : 17
Date solved : Thursday, October 02, 2025 at 09:27:31 PM
CAS classification : [_separable]

\begin{align*} y^{\prime }&=\frac {y \left (b_{2} x +b_{1} \right )}{x \left (a_{1} +a_{2} y\right )} \end{align*}

✓ Maple. Time used: 0.009 (sec). Leaf size: 48

ode:=diff(y(x),x) = y(x)*(b__2*x+b__1)/x/(a__1+a__2*y(x)); 
dsolve(ode,y(x), singsol=all);
 

\[ y = x^{\frac {b_{1}}{a_{1}}} {\mathrm e}^{\frac {-a_{1} \operatorname {LambertW}\left (\frac {a_{2} x^{\frac {b_{1}}{a_{1}}} {\mathrm e}^{\frac {b_{2} x +c_1}{a_{1}}}}{a_{1}}\right )+b_{2} x +c_1}{a_{1}}} \]

✓ Mathematica. Time used: 1.474 (sec). Leaf size: 40

ode=D[y[x],x]== y[x]*(b1+b2*x)/( x*(a1+a2*y[x]) ); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {\text {a1} W\left (\frac {\text {a2} x^{\frac {\text {b1}}{\text {a1}}} e^{\frac {\text {b2} x+c_1}{\text {a1}}}}{\text {a1}}\right )}{\text {a2}}\\ y(x)&\to 0 \end{align*}

✓ Sympy. Time used: 0.297 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
b1 = symbols("b1") 
b2 = symbols("b2") 
a1 = symbols("a1") 
a2 = symbols("a2") 
y = Function("y") 
ode = Eq(Derivative(y(x), x) - (b1 + b2*x)*y(x)/(x*(a1 + a2*y(x))),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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