80.3.7 problem 8

Internal problem ID [18472]
Book : Elementary Differential Equations. By Thornton C. Fry. D Van Nostrand. NY. First Edition (1929)
Section : Chapter IV. Methods of solution: First order equations. section 29. Problems at page 81
Problem number : 8
Date solved : Friday, March 14, 2025 at 04:54:14 AM
CAS classification : [_exact]

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

✓ Maple. Time used: 0.006 (sec). Leaf size: 79

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

\[ \frac {\left (\frac {t^{2}}{2}+\frac {T^{2}}{2}+c_{1} \right ) \sqrt {-T^{2}}+T \left (\ln \left (\frac {\sqrt {-T^{2}}\, \sqrt {t^{2}-T^{2}}-T^{2}}{t}\right )+\ln \left (2\right )-\ln \left (T\right )\right )}{\sqrt {-T^{2}}} = 0 \]

✓ Mathematica. Time used: 1.592 (sec). Leaf size: 44

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

\[ \text {Solve}\left [-\arctan \left (\frac {\sqrt {t^2-T(t)^2}}{T(t)}\right )+\frac {t^2}{2}+\frac {T(t)^2}{2}=c_1,T(t)\right ] \]

✗ Sympy

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

Timed Out