74.8.15 problem 15

Internal problem ID [16151]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 2. First Order Equations. Review exercises, page 80
Problem number : 15
Date solved : Tuesday, January 28, 2025 at 08:52:24 AM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

\begin{align*} x^{\prime }&=\frac {5 t x}{t^{2}+x^{2}} \end{align*}

✓ Solution by Maple

Time used: 8.095 (sec). Leaf size: 522

dsolve(diff(x(t),t)=5*t*x(t)/(x(t)^2+t^2),x(t), singsol=all)
 

\[ x \left (t \right ) = \frac {4 \operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{72}-4 \operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{64}+\operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{56}+1024 \operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{32} t^{8} c_{1} -1280 \operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{24} t^{8} c_{1} +640 \operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{16} t^{8} c_{1} -160 \operatorname {RootOf}\left (4 \textit {\_Z}^{80}-4 \textit {\_Z}^{72}+\textit {\_Z}^{64}+1024 \textit {\_Z}^{40} c_{1} t^{8}-1280 \textit {\_Z}^{32} c_{1} t^{8}+640 \textit {\_Z}^{24} c_{1} t^{8}-160 \textit {\_Z}^{16} c_{1} t^{8}+20 \textit {\_Z}^{8} c_{1} t^{8}-c_{1} t^{8}\right )^{8} t^{8} c_{1} +18 c_{1} t^{8}}{c_{1} t^{7}} \]

✓ Solution by Mathematica

Time used: 0.138 (sec). Leaf size: 46

DSolve[D[x[t],t]==5*t*x[t]/(x[t]^2+t^2),x[t],t,IncludeSingularSolutions -> True]
 

\[ \text {Solve}\left [\int _1^{\frac {x(t)}{t}}\frac {K[1]^2+1}{(K[1]-2) K[1] (K[1]+2)}dK[1]=-\log (t)+c_1,x(t)\right ] \]