90.5.8 problem 8

Internal problem ID [25127]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 1. First order differential equations. Exercises at page 71
Problem number : 8
Date solved : Thursday, October 02, 2025 at 11:53:02 PM
CAS classification : [[_homogeneous, `class A`], _dAlembert]

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

✓ Maple. Time used: 0.032 (sec). Leaf size: 29

ode:=t^2*diff(y(t),t) = t*y(t)+y(t)*(t^2+y(t)^2)^(1/2); 
dsolve(ode,y(t), singsol=all);
 

\[ \frac {t \sqrt {t^{2}+y^{2}}-c_1 y+t^{2}}{y} = 0 \]

✓ Mathematica. Time used: 0.208 (sec). Leaf size: 47

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

\begin{align*} y(t)&\to -t \sqrt {-\text {sech}^2(\log (t)+c_1)}\\ y(t)&\to t \sqrt {-\text {sech}^2(\log (t)+c_1)}\\ y(t)&\to 0 \end{align*}

✓ Sympy. Time used: 1.271 (sec). Leaf size: 10

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

\[ y{\left (t \right )} = \frac {t}{\sinh {\left (C_{1} - \log {\left (t \right )} \right )}} \]