90.35.5 problem 5

Internal problem ID [25506]
Book : Ordinary Differential Equations. By William Adkins and Mark G Davidson. Springer. NY. 2010. ISBN 978-1-4614-3617-1
Section : Chapter 9. Linear Systems of Differential Equations. Exercises at page 719
Problem number : 5
Date solved : Sunday, October 12, 2025 at 05:55:40 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} y_{1} \left (\pi \right )&=1 \\ y_{2} \left (\pi \right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.070 (sec). Leaf size: 41

ode:=[diff(y__1(t),t) = 1/t*y__1(t)+y__2(t), diff(y__2(t),t) = -y__1(t)+1/t*y__2(t)]; 
ic:=[y__1(Pi) = 1, y__2(Pi) = -1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= t \left (\frac {\sin \left (t \right )}{\pi }-\frac {\cos \left (t \right )}{\pi }\right ) \\ y_{2} \left (t \right ) &= -t \left (-\frac {\sin \left (t \right )}{\pi }-\frac {\cos \left (t \right )}{\pi }\right ) \\ \end{align*}

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 30

ode={D[y1[t],t]==1/t*y1[t]+y2[t], D[y2[t],t]==-y1[t]+1/t*y2[t]}; 
ic={y1[Pi]==1,y2[Pi]==-1}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to \frac {t (\sin (t)-\cos (t))}{\pi }\\ \text {y2}(t)&\to \frac {t (\sin (t)+\cos (t))}{\pi } \end{align*}

✓ Sympy. Time used: 0.196 (sec). Leaf size: 63

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-y2(t) + Derivative(y1(t), t) - y1(t)/t,0),Eq(y1(t) + Derivative(y2(t), t) - y2(t)/t,0)] 
ics = {y1(pi): 1, y2(pi): -1} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = - \frac {i t \left (1 - i\right ) e^{i t}}{2 \pi } - \frac {t \left (1 - i\right ) e^{- i t}}{2 \pi }, \ y_{2}{\left (t \right )} = \frac {t \left (1 - i\right ) e^{i t}}{2 \pi } + \frac {i t \left (1 - i\right ) e^{- i t}}{2 \pi }\right ] \]