87.19.3 problem 3

Internal problem ID [23673]
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 : 3
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 )&=2 \sin \left (t \right ) x_{1} \left (t \right )+\ln \left (t \right ) x_{2} \left (t \right )\\ \frac {d}{d t}x_{2} \left (t \right )&=\frac {x_{1} \left (t \right )}{t -2}+\frac {{\mathrm e}^{t} x_{2} \left (t \right )}{1+t} \end{align*}

With initial conditions

\begin{align*} x_{1} \left (3\right )&=0 \\ x_{2} \left (3\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 9

ode:=[diff(x__1(t),t) = 2*sin(t)*x__1(t)+ln(t)*x__2(t), diff(x__2(t),t) = 1/(t-2)*x__1(t)+exp(t)/(t+1)*x__2(t)]; 
ic:=[x__1(3) = 0, x__2(3) = 0]; 
dsolve([ode,op(ic)]);
 

\begin{align*} x_{1} \left (t \right ) &= 0 \\ x_{2} \left (t \right ) &= 0 \\ \end{align*}

✗ Mathematica

ode={D[x1[t],t]==2*Sin[t]*x1[t]+Log[t]*x2[t],D[x2[t],t]==1/(t-2)*x1[t]+Exp[t]/(1+t)*x2[t]}; 
ic={x1[3]==0,x2[3]==0}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

Not solved

✓ Sympy. Time used: 0.425 (sec). Leaf size: 7

from sympy import * 
t = symbols("t") 
x1 = Function("x1") 
x2 = Function("x2") 
ode=[Eq(-2*x1(t)*sin(t) - x2(t)*log(t) + Derivative(x1(t), t),0),Eq(Derivative(x2(t), t) - x2(t)*exp(t)/(t + 1) - x1(t)/(t - 2),0)] 
ics = {x1(3): 0, x2(3): 0} 
dsolve(ode,func=[x1(t),x2(t)],ics=ics)
 

\[ \left [ x_{1}{\left (t \right )} = 0, \ x_{2}{\left (t \right )} = 0\right ] \]