87.19.12 problem 17

Internal problem ID [23682]
Book : Ordinary differential equations with modern applications. Ladas, G. E. and Finizio, N. Wadsworth Publishing. California. 1978. ISBN 0-534-00552-7. QA372.F56
Section : Chapter 3. Linear Systems. Exercise at page 149
Problem number : 17
Date solved : Sunday, October 12, 2025 at 05:55:14 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=x_{1} \left (t \right )+\left (1-t \right ) x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=\frac {x_{1} \left (t \right )}{t}-x_{2} \left (t \right ) \end{align*}

✓ Maple. Time used: 0.024 (sec). Leaf size: 23

ode:=[diff(x__1(t),t) = x__1(t)+(1-t)*x__2(t), diff(x__2(t),t) = 1/t*x__1(t)-x__2(t)]; 
dsolve(ode);
 

\begin{align*} x_{1} \left (t \right ) &= \ln \left (t \right ) c_2 t +c_1 t +c_2 \\ x_{2} \left (t \right ) &= c_2 \ln \left (t \right )+c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.048 (sec). Leaf size: 32

ode={D[x1[t],t]==x1[t]+(1-t)*x2[t],D[x2[t],t]==1/t*x1[t]-x2[t]}; 
ic={}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)&\to c_1 t-c_2 (t \log (t)+1)\\ \text {x2}(t)&\to c_1-c_2 \log (t) \end{align*}

✓ Sympy. Time used: 0.115 (sec). Leaf size: 83

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(-(1 - t)*x2(t) - x1(t) + Derivative(x1(t), t),0),Eq(x2(t) + Derivative(x2(t), t) - x1(t)/t,0)] 
ics = {} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

\[ \left [ x_{1}{\left (t \right )} = C_{1} x_{0}{\left (t \right )} + C_{2} x_{0}{\left (t \right )} \int \frac {\left (1 - t\right ) \left (e^{\int \left (-1\right )\, dt}\right ) e^{\int 1\, dt}}{x_{0}^{2}{\left (t \right )}}\, dt, \ x_{2}{\left (t \right )} = C_{1} y_{0}{\left (t \right )} + C_{2} \left (y_{0}{\left (t \right )} \int \frac {\left (1 - t\right ) \left (e^{\int \left (-1\right )\, dt}\right ) e^{\int 1\, dt}}{x_{0}^{2}{\left (t \right )}}\, dt + \frac {\left (e^{\int \left (-1\right )\, dt}\right ) e^{\int 1\, dt}}{x_{0}{\left (t \right )}}\right )\right ] \]