90.35.3 problem 3

Internal problem ID [25504]
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 : 3
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_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-t y_{1} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.057 (sec). Leaf size: 21

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

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

✓ Mathematica. Time used: 0.002 (sec). Leaf size: 26

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

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

✓ Sympy. Time used: 0.164 (sec). Leaf size: 49

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

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