68.8.15 problem 15

Internal problem ID [17438]
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 : Thursday, October 02, 2025 at 02:21:55 PM
CAS classification : [[_homogeneous, `class A`], _rational, _dAlembert]

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

✓ Maple. Time used: 2.652 (sec). Leaf size: 522

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

\begin{align*} \text {Solution too large to show}\end{align*}

✓ Mathematica. Time used: 60.08 (sec). Leaf size: 641

ode=D[x[t],t]==5*t*x[t]/(x[t]^2+t^2); 
ic={}; 
DSolve[{ode,ic},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*}

✓ Sympy. Time used: 0.365 (sec). Leaf size: 22

from sympy import * 
t = symbols("t") 
x = Function("x") 
ode = Eq(-5*t*x(t)/(t**2 + x(t)**2) + Derivative(x(t), t),0) 
ics = {} 
dsolve(ode,func=x(t),ics=ics)
 

\[ \log {\left (x{\left (t \right )} \right )} = C_{1} - \log {\left (\left (\frac {4 t^{2}}{x^{2}{\left (t \right )}} - 1\right )^{\frac {5}{8}} \right )} \]