74.7.40 problem 40

Internal problem ID [16046]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Exercises 2.5, page 64
Problem number : 40
Date solved : Monday, March 31, 2025 at 02:36:13 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} t y^{3}-\left (t^{4}+y^{4}\right ) y^{\prime }&=0 \end{align*}

With initial conditions

\begin{align*} y \left (1\right )&=1 \end{align*}

✓ Maple. Time used: 0.517 (sec). Leaf size: 56

ode:=t*y(t)^3-(t^4+y(t)^4)*diff(y(t),t) = 0; 
ic:=y(1) = 1; 
dsolve([ode,ic],y(t), singsol=all);
 

\[ y = {\mathrm e}^{\frac {\sqrt {3}\, \left (6 \operatorname {RootOf}\left (-3 \tan \left (\textit {\_Z} \right ) t^{2} {\mathrm e}^{-\frac {\sqrt {3}\, \pi }{9}}+\sqrt {3}\, t^{2} {\mathrm e}^{-\frac {\sqrt {3}\, \pi }{9}}-2 \,{\mathrm e}^{\frac {2 \sqrt {3}\, \textit {\_Z}}{3}} \sqrt {3}\right )+\pi \right )}{18}} \]

✓ Mathematica. Time used: 0.126 (sec). Leaf size: 45

ode=(t*y[t]^3)-(t^4+y[t]^4)*D[y[t],t]==0; 
ic={y[1]==1}; 
DSolve[{ode,ic},y[t],t,IncludeSingularSolutions->True]
 

\[ \text {Solve}\left [\int _1^{\frac {y(t)}{t}}\frac {K[1]^4+1}{K[1] \left (K[1]^4-K[1]^2+1\right )}dK[1]=-\log (t),y(t)\right ] \]

✓ Sympy. Time used: 1.079 (sec). Leaf size: 39

from sympy import * 
t = symbols("t") 
y = Function("y") 
ode = Eq(t*y(t)**3 - (t**4 + y(t)**4)*Derivative(y(t), t),0) 
ics = {y(1): 1} 
dsolve(ode,func=y(t),ics=ics)
 

\[ \log {\left (y{\left (t \right )} \right )} = \frac {\sqrt {3} \operatorname {atan}{\left (\frac {\sqrt {3} \left (\frac {2 t^{2}}{y^{2}{\left (t \right )}} - 1\right )}{3} \right )}}{3} - \frac {\sqrt {3} \pi }{18} \]