83.8.22 problem 23

Internal problem ID [19077]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter II. Equations of first order and first degree. Misc examples on chapter II at page 25
Problem number : 23
Date solved : Monday, March 31, 2025 at 06:45:00 PM
CAS classification : [_separable]

\begin{align*} s^{\prime }+x^{2}&=x^{2} {\mathrm e}^{3 s} \end{align*}

✓ Maple. Time used: 0.014 (sec). Leaf size: 19

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

\[ s = \frac {\ln \left (-\frac {1}{{\mathrm e}^{x^{3}} c_1 -1}\right )}{3} \]

✓ Mathematica. Time used: 0.782 (sec). Leaf size: 132

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

\begin{align*} s(x)\to \log \left (-\sqrt [3]{-\frac {1}{2}} \sqrt [3]{1-\tanh \left (\frac {1}{2} \left (x^3+3 c_1\right )\right )}\right ) \\ s(x)\to \log \left (\frac {\sqrt [3]{1-\tanh \left (\frac {1}{2} \left (x^3+3 c_1\right )\right )}}{\sqrt [3]{2}}\right ) \\ s(x)\to \log \left (\frac {(-1)^{2/3} \sqrt [3]{1-\tanh \left (\frac {1}{2} \left (x^3+3 c_1\right )\right )}}{\sqrt [3]{2}}\right ) \\ s(x)\to 0 \\ s(x)\to -\frac {2 i \pi }{3} \\ s(x)\to \frac {2 i \pi }{3} \\ \end{align*}

✓ Sympy. Time used: 0.397 (sec). Leaf size: 20

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

\[ - \frac {x^{3}}{3} - s{\left (x \right )} + \frac {\log {\left (e^{3 s{\left (x \right )}} - 1 \right )}}{3} = C_{1} \]