76.3.18 problem 18

Internal problem ID [17318]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 2. First order differential equations. Section 2.4 (Differences between linear and nonlinear equations). Problems at page 79
Problem number : 18
Date solved : Monday, March 31, 2025 at 03:52:28 PM
CAS classification : [_separable]

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

With initial conditions

\begin{align*} y \left (0\right )&=y_{0} \end{align*}

✗ Maple

ode:=diff(y(t),t) = t^2/y(t)/(t^3+1); 
ic:=y(0) = y__0; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ \text {No solution found} \]

✓ Mathematica. Time used: 0.107 (sec). Leaf size: 47

ode=D[y[t],t]==t^2/(y[t]*(1+t^3)); 
ic={y[0]==y0}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\begin{align*} y(t)\to -\sqrt {\frac {2}{3} \log \left (t^3+1\right )+\text {y0}^2} \\ y(t)\to \sqrt {\frac {2}{3} \log \left (t^3+1\right )+\text {y0}^2} \\ \end{align*}

✓ Sympy. Time used: 0.380 (sec). Leaf size: 20

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(-t**2/((t**3 + 1)*y(t)) + Derivative(y(t), t),0) 
ics = {y(0): y__0} 
dsolve(ode,func=y(t),ics=ics)
 

\[ y{\left (t \right )} = \frac {\sqrt {9 \left (y^{0}\right )^{2} + 6 \log {\left (t^{3} + 1 \right )}}}{3} \]