75.3.17 problem 17

Internal problem ID [19921]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter III. Ordinary differential equations of the first order and first degree. Exercises at page 33
Problem number : 17
Date solved : Thursday, October 02, 2025 at 05:01:25 PM
CAS classification : [_separable]

\begin{align*} y \left (3+y\right ) y^{\prime }&=x \left (3+2 y\right ) \end{align*}

✓ Maple. Time used: 0.077 (sec). Leaf size: 50

ode:=y(x)*(3+y(x))*diff(y(x),x) = x*(2*y(x)+3); 
dsolve(ode,y(x), singsol=all);
 

\[ y = \frac {3 \,{\mathrm e}^{-\frac {1}{2}-\frac {4 x^{2}}{9}-\frac {8 c_1}{9}}}{2 \sqrt {-\frac {{\mathrm e}^{-1-\frac {8 x^{2}}{9}-\frac {16 c_1}{9}}}{\operatorname {LambertW}\left (-\frac {{\mathrm e}^{-1-\frac {8 x^{2}}{9}-\frac {16 c_1}{9}}}{9}\right )}}}-\frac {3}{2} \]

✓ Mathematica. Time used: 0.219 (sec). Leaf size: 48

ode=y[x]*(3+y[x])*D[y[x],x]==x*(2*y[x]+3); 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \text {InverseFunction}\left [\frac {1}{16} (2 \text {$\#$1}+3)^2-\frac {9}{8} \log (2 \text {$\#$1}+3)\&\right ]\left [\frac {x^2}{2}+c_1\right ]\\ y(x)&\to -\frac {3}{2} \end{align*}

✓ Sympy. Time used: 0.306 (sec). Leaf size: 31

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*(2*y(x) + 3) + (y(x) + 3)*y(x)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

\[ - \frac {x^{2}}{2} + \frac {y^{2}{\left (x \right )}}{4} + \frac {3 y{\left (x \right )}}{4} - \frac {9 \log {\left (2 y{\left (x \right )} + 3 \right )}}{8} = C_{1} \]