68.7.38 problem 38

Internal problem ID [17402]
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 : 38
Date solved : Thursday, October 02, 2025 at 02:17:03 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

\begin{align*} t^{3}+y^{2} \sqrt {t^{2}+y^{2}}-t y \sqrt {t^{2}+y^{2}}\, y^{\prime }&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.379 (sec). Leaf size: 22

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

\[ y = \sqrt {-1+\left (2 \sqrt {2}+3 \ln \left (t \right )\right )^{{2}/{3}}}\, t \]

✓ Mathematica. Time used: 24.367 (sec). Leaf size: 80

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

\begin{align*} y(t)&\to \sqrt {\sqrt [3]{-t^6 \left (-9 \log ^2(t)+12 \sqrt {2} \log (t)-8\right )}-t^2}\\ y(t)&\to \sqrt {\sqrt [3]{t^6 \left (9 \log ^2(t)+12 \sqrt {2} \log (t)+8\right )}-t^2} \end{align*}

✗ Sympy

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

TypeError : cannot determine truth value of Relational: 18*C1*t**6 < 18*t**6