80.3.7 problem 8

Internal problem ID [18551]
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 : Tuesday, January 28, 2025 at 11:58:04 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*}

✓ Solution by Maple

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

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

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

✓ Solution by Mathematica

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

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

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