Internal problem ID [14417]
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.
ODE order: 1.
ODE degree: 1.
CAS Maple gives this as type [[_homogeneous, `class A`], _rational, _dAlembert]
\[ \boxed {x^{\prime }-\frac {5 t x}{x^{2}+t^{2}}=0} \]
✓ Solution by Maple
Time used: 14.282 (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: 60.117 (sec). Leaf size: 641
DSolve[x'[t]==5*t*x[t]/(x[t]^2+t^2),x[t],t,IncludeSingularSolutions -> True]
\begin{align*} x(t)\to -\sqrt {\text {Root}\left [\text {$\#$1}^5-20 \text {$\#$1}^4 t^2+160 \text {$\#$1}^3 t^4-640 \text {$\#$1}^2 t^6+\text {$\#$1} \left (1280 t^8+e^{8 c_1}\right )-1024 t^{10}\&,1\right ]} \\ x(t)\to \sqrt {\text {Root}\left [\text {$\#$1}^5-20 \text {$\#$1}^4 t^2+160 \text {$\#$1}^3 t^4-640 \text {$\#$1}^2 t^6+\text {$\#$1} \left (1280 t^8+e^{8 c_1}\right )-1024 t^{10}\&,1\right ]} \\ x(t)\to -\sqrt {\text {Root}\left [\text {$\#$1}^5-20 \text {$\#$1}^4 t^2+160 \text {$\#$1}^3 t^4-640 \text {$\#$1}^2 t^6+\text {$\#$1} \left (1280 t^8+e^{8 c_1}\right )-1024 t^{10}\&,2\right ]} \\ x(t)\to \sqrt {\text {Root}\left [\text {$\#$1}^5-20 \text {$\#$1}^4 t^2+160 \text {$\#$1}^3 t^4-640 \text {$\#$1}^2 t^6+\text {$\#$1} \left (1280 t^8+e^{8 c_1}\right )-1024 t^{10}\&,2\right ]} \\ x(t)\to -\sqrt {\text {Root}\left [\text {$\#$1}^5-20 \text {$\#$1}^4 t^2+160 \text {$\#$1}^3 t^4-640 \text {$\#$1}^2 t^6+\text {$\#$1} \left (1280 t^8+e^{8 c_1}\right )-1024 t^{10}\&,3\right ]} \\ x(t)\to \sqrt {\text {Root}\left [\text {$\#$1}^5-20 \text {$\#$1}^4 t^2+160 \text {$\#$1}^3 t^4-640 \text {$\#$1}^2 t^6+\text {$\#$1} \left (1280 t^8+e^{8 c_1}\right )-1024 t^{10}\&,3\right ]} \\ x(t)\to -\sqrt {\text {Root}\left [\text {$\#$1}^5-20 \text {$\#$1}^4 t^2+160 \text {$\#$1}^3 t^4-640 \text {$\#$1}^2 t^6+\text {$\#$1} \left (1280 t^8+e^{8 c_1}\right )-1024 t^{10}\&,4\right ]} \\ x(t)\to \sqrt {\text {Root}\left [\text {$\#$1}^5-20 \text {$\#$1}^4 t^2+160 \text {$\#$1}^3 t^4-640 \text {$\#$1}^2 t^6+\text {$\#$1} \left (1280 t^8+e^{8 c_1}\right )-1024 t^{10}\&,4\right ]} \\ x(t)\to -\sqrt {\text {Root}\left [\text {$\#$1}^5-20 \text {$\#$1}^4 t^2+160 \text {$\#$1}^3 t^4-640 \text {$\#$1}^2 t^6+\text {$\#$1} \left (1280 t^8+e^{8 c_1}\right )-1024 t^{10}\&,5\right ]} \\ x(t)\to \sqrt {\text {Root}\left [\text {$\#$1}^5-20 \text {$\#$1}^4 t^2+160 \text {$\#$1}^3 t^4-640 \text {$\#$1}^2 t^6+\text {$\#$1} \left (1280 t^8+e^{8 c_1}\right )-1024 t^{10}\&,5\right ]} \\ \end{align*}