90.35.8 problem 8

Internal problem ID [25509]
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 : 8
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_{1} \left (t \right )}{t}+1\\ \frac {d}{d t}y_{2} \left (t \right )&=\frac {y_{2} \left (t \right )}{t}+t \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.025 (sec). Leaf size: 18

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

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

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

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

\begin{align*} \text {y1}(t)&\to t (\log (t)+1)\\ \text {y2}(t)&\to t (t+1) \end{align*}

✗ Sympy

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

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