90.35.4 problem 4

Internal problem ID [25505]
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 : 4
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 )&=t y_{1} \left (t \right )+t y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-t y_{1} \left (t \right )-t y_{2} \left (t \right ) \end{align*}

With initial conditions

\begin{align*} y_{1} \left (0\right )&=4 \\ y_{2} \left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 19

ode:=[diff(y__1(t),t) = t*y__1(t)+t*y__2(t), diff(y__2(t),t) = -t*y__1(t)-t*y__2(t)]; 
ic:=[y__1(0) = 4, y__2(0) = 0]; 
dsolve([ode,op(ic)]);
 

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

✓ Mathematica. Time used: 0.029 (sec). Leaf size: 20

ode={D[y1[t],t]==t*y1[t]+t*y2[t], D[y2[t],t]==-t*y1[t]-t*y2[t]}; 
ic={y1[0]==4,y2[0]==0}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(t)&\to 2 \left (t^2+2\right )\\ \text {y2}(t)&\to -2 t^2 \end{align*}

✓ Sympy. Time used: 0.120 (sec). Leaf size: 17

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

\[ \left [ y_{1}{\left (t \right )} = 2 t^{2} + 4, \ y_{2}{\left (t \right )} = - 2 t^{2}\right ] \]