90.33.3 problem 3

Internal problem ID [25477]
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 645
Problem number : 3
Date solved : Sunday, October 12, 2025 at 05:55:39 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.226 (sec). Leaf size: 34

ode:=[diff(y__1(t),t) = sin(t)*y__1(t), diff(y__2(t),t) = y__1(t)+cos(t)*y__2(t)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (t \right ) &= c_2 \,{\mathrm e}^{-\cos \left (t \right )} \\ y_{2} \left (t \right ) &= {\mathrm e}^{\sin \left (t \right )} \left (\int c_2 \,{\mathrm e}^{-\cos \left (t \right )-\sin \left (t \right )}d t +c_1 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.641 (sec). Leaf size: 48

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

\begin{align*} \text {y1}(t)&\to c_1 e^{-\cos (t)}\\ \text {y2}(t)&\to e^{\sin (t)} \left (c_1 \int _1^te^{-\cos (K[1])-\sin (K[1])}dK[1]+c_2\right ) \end{align*}

✓ Sympy. Time used: 34.245 (sec). Leaf size: 37

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

\[ \left [ y_{1}{\left (t \right )} = C_{1} e^{- \cos {\left (t \right )}}, \ y_{2}{\left (t \right )} = C_{1} e^{\sin {\left (t \right )}} \int e^{- \sin {\left (t \right )}} e^{- \cos {\left (t \right )}}\, dt + C_{2} e^{\sin {\left (t \right )}}\right ] \]