90.35.7 problem 7

Internal problem ID [25508]
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 : 7
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 )&=y_{1} \left (t \right )+y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=\frac {y_{1} \left (t \right )}{t}+\frac {y_{2} \left (t \right )}{t} \end{align*}

With initial conditions

\begin{align*} y_{1} \left (1\right )&=-3 \\ y_{2} \left (1\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.053 (sec). Leaf size: 28

ode:=[diff(y__1(t),t) = y__1(t)+y__2(t), diff(y__2(t),t) = 1/t*y__1(t)+1/t*y__2(t)]; 
ic:=[y__1(1) = -3, y__2(1) = 4]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= -3+\left (t -1\right ) {\mathrm e}^{t} {\mathrm e}^{-1} \\ y_{2} \left (t \right ) &= {\mathrm e}^{t} {\mathrm e}^{-1}+3 \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 26

ode={D[y1[t],t]==y1[t]+y2[t], D[y2[t],t]==1/t*y1[t]+1/t*y2[t]}; 
ic={y1[1]==-3,y2[1]==4}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to e^{t-1} (t-1)-3\\ \text {y2}(t)&\to e^{t-1}+3 \end{align*}

✓ Sympy. Time used: 1.268 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-y1(t) - y2(t) + Derivative(y1(t), t),0),Eq(Derivative(y2(t), t) - y1(t)/t - y2(t)/t,0)] 
ics = {y1(1): -3, y2(1): 4} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = - C_{3} + t^{\operatorname {re}{\left (t\right )} + 2} \left (C_{2} \sin {\left (\log {\left (t \right )} \left |{\operatorname {im}{\left (t\right )}}\right | \right )} + C_{3} \cos {\left (\log {\left (t \right )} \operatorname {im}{\left (t\right )} \right )}\right ) - 3, \ y_{2}{\left (t \right )} = 4\right ] \]