25.1.21 problem 10.3

Internal problem ID [6854]
Book : Differential Equations, By George Boole F.R.S. 1865
Section : Chapter 2
Problem number : 10.3
Date solved : Tuesday, September 30, 2025 at 03:55:55 PM
CAS classification : [_Bernoulli]

\begin{align*} z^{\prime }+2 x z&=2 a \,x^{3} z^{3} \end{align*}

✓ Maple. Time used: 0.003 (sec). Leaf size: 53

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

\begin{align*} z &= -\frac {2}{\sqrt {4 a \,x^{2}+4 \,{\mathrm e}^{2 x^{2}} c_1 +2 a}} \\ z &= \frac {2}{\sqrt {4 a \,x^{2}+4 \,{\mathrm e}^{2 x^{2}} c_1 +2 a}} \\ \end{align*}

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 29

ode=D[z[x],x]+2*x*z[x]==2*a*x^3*z[x]; 
ic={}; 
DSolve[{ode,ic},z[x],x,IncludeSingularSolutions->True]
 

\begin{align*} z(x)&\to c_1 e^{\frac {1}{2} x^2 \left (a x^2-2\right )}\\ z(x)&\to 0 \end{align*}

✓ Sympy. Time used: 1.009 (sec). Leaf size: 60

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

\[ \left [ z{\left (x \right )} = - \sqrt {2} \sqrt {\frac {1}{C_{1} e^{2 x^{2}} + 2 a x^{2} + a}}, \ z{\left (x \right )} = \sqrt {2} \sqrt {\frac {1}{C_{1} e^{2 x^{2}} + 2 a x^{2} + a}}\right ] \]