19.1.17 problem 17

Internal problem ID [4229]
Book : Advanced Mathematica, Book2, Perkin and Perkin, 1992
Section : Chapter 11.3, page 316
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 07:07:52 AM
CAS classification : [_quadrature]

\begin{align*} \left (x^{3}+1\right ) y^{\prime }&=3 x^{2} \tan \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=\frac {\pi }{2} \\ \end{align*}

✓ Maple. Time used: 0.048 (sec). Leaf size: 37

ode:=(x^3+1)*diff(y(x),x) = 3*x^2*tan(x); 
ic:=[y(0) = 1/2*Pi]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = 3 \int _{0}^{x}\frac {\textit {\_z1}^{2} \tan \left (\textit {\_z1} \right )}{\left (\textit {\_z1} +1\right ) \left (\textit {\_z1}^{2}-\textit {\_z1} +1\right )}d \textit {\_z1} +\frac {\pi }{2} \]

✓ Mathematica. Time used: 5.499 (sec). Leaf size: 35

ode=(1+x^3)*D[y[x],x]==3*x^2*Tan[x]; 
ic=y[0]==Pi/2; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \int _0^x\frac {3 K[1]^2 \tan (K[1])}{K[1]^3+1}dK[1]+\frac {\pi }{2} \end{align*}

✓ Sympy. Time used: 0.298 (sec). Leaf size: 41

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*x**2*tan(x) + (x**3 + 1)*Derivative(y(x), x),0) 
ics = {y(0): pi/2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = - 3 \int \limits ^{0} \frac {x^{2} \tan {\left (x \right )}}{x^{3} + 1}\, dx + 3 \int \frac {x^{2} \tan {\left (x \right )}}{\left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx + \frac {\pi }{2} \]