90.35.9 problem 9

Internal problem ID [25510]
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 : 9
Date solved : Sunday, October 12, 2025 at 05:55:41 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.026 (sec). Leaf size: 19

ode:=[diff(y__1(t),t) = -1/t*y__2(t)+1, diff(y__2(t),t) = 1/t*y__1(t)+2/t*y__2(t)-1]; 
ic:=[y__1(1) = 2, y__2(1) = 0]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 19

ode={D[y1[t],t]==-1/t*y2[t]+1, D[y2[t],t]==1/t*y1[t]+2/t*y2[t]-1}; 
ic={y1[1]==2,y2[1]==0}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to -t (\log (t)-2)\\ \text {y2}(t)&\to t \log (t) \end{align*}

✗ Sympy

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(Derivative(y1(t), t) - 1 + y2(t)/t,0),Eq(Derivative(y2(t), t) + 1 - y1(t)/t - 2*y2(t)/t,0)] 
ics = {y1(1): 2, y2(1): 0} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

NotAlgebraic : Integral(-1, (t_, 0)) does not seem to be an algebraic element