80.5.4 problem 5

Internal problem ID [18562]
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 32. Problems at page 89
Problem number : 5
Date solved : Tuesday, January 28, 2025 at 12:00:08 PM
CAS classification : [[_homogeneous, `class G`]]

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 136

dsolve(sqrt(t^2+T(t))=diff(T(t),t),T(t), singsol=all)
 

\[ 17 \ln \left (-t^{4}-t^{2} T+4 T^{2}\right )+17 \ln \left (-\sqrt {t^{2}+T}\, t +2 T\right )-17 \ln \left (\sqrt {t^{2}+T}\, t +2 T\right )+\left (2 \,\operatorname {arctanh}\left (\frac {\left (t^{2}-8 T\right ) \sqrt {17}}{17 t^{2}}\right )+2 \,\operatorname {arctanh}\left (\frac {\left (t -4 \sqrt {t^{2}+T}\right ) \sqrt {17}}{17 t}\right )-2 \,\operatorname {arctanh}\left (\frac {\left (4 \sqrt {t^{2}+T}+t \right ) \sqrt {17}}{17 t}\right )\right ) \sqrt {17}-c_{1} = 0 \]

✓ Solution by Mathematica

Time used: 0.277 (sec). Leaf size: 135

DSolve[Sqrt[t^2+T[t]]==D[T[t],t],T[t],t,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\frac {1}{34} \left (-34 \log \left (\sqrt {t^2+T(t)}-t\right )-\left (\sqrt {17}-17\right ) \log \left (2 \left (\sqrt {17}-4\right ) t \sqrt {t^2+T(t)}-2 \left (\sqrt {17}-4\right ) t^2-\left (\sqrt {17}-3\right ) T(t)\right )+\left (17+\sqrt {17}\right ) \log \left (2 \left (4+\sqrt {17}\right ) t \sqrt {t^2+T(t)}-2 \left (4+\sqrt {17}\right ) t^2-\left (3+\sqrt {17}\right ) T(t)\right )\right )=c_1,T(t)\right ] \]