90.35.11 problem 11

Internal problem ID [25512]
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 : 11
Date solved : Sunday, October 12, 2025 at 05:55:42 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}y_{1} \left (t \right )&=3 \sec \left (t \right ) y_{1} \left (t \right )+5 \sec \left (t \right ) y_{2} \left (t \right )\\ \frac {d}{d t}y_{2} \left (t \right )&=-\sec \left (t \right ) y_{1} \left (t \right )-3 \sec \left (t \right ) y_{2} \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.177 (sec). Leaf size: 93

ode:=[diff(y__1(t),t) = 3*sec(t)*y__1(t)+5*sec(t)*y__2(t), diff(y__2(t),t) = -sec(t)*y__1(t)-3*sec(t)*y__2(t)]; 
ic:=[y__1(0) = 2, y__2(0) = 1]; 
dsolve([ode,op(ic)]);
 

\begin{align*} y_{1} \left (t \right ) &= -\frac {7}{4 \left (\sec \left (t \right )+\tan \left (t \right )\right )^{2}}+\frac {15 \left (\sec \left (t \right )+\tan \left (t \right )\right )^{2}}{4} \\ y_{2} \left (t \right ) &= \frac {-3 \sec \left (t \right ) \left (-\frac {7}{4}+\frac {15 \left (\sec \left (t \right )+\tan \left (t \right )\right )^{4}}{4}\right ) \left (\sec \left (t \right )+\tan \left (t \right )\right )+\frac {7 \sec \left (t \right ) \tan \left (t \right )}{2}+\frac {7 \tan \left (t \right )^{2}}{2}+\frac {7}{2}+\frac {15 \left (\sec \left (t \right )+\tan \left (t \right )\right )^{4} \left (\sec \left (t \right ) \tan \left (t \right )+\tan \left (t \right )^{2}+1\right )}{2}}{5 \left (\sec \left (t \right )+\tan \left (t \right )\right )^{3} \sec \left (t \right )} \\ \end{align*}

✗ Mathematica

ode={D[y1[t],t]==3*Sec[t]*y1[t]+5*Sec[t]*y2[t], D[y2[t],t]==-Sec[t]*y1[t]-3*Sec[t]*y2[t]}; 
ic={y1[0]==2,y2[0]==1}; 
DSolve[{ode,ic},{y1[t],y2[t]},t,IncludeSingularSolutions->True]
 

Not solved

✓ Sympy. Time used: 2.493 (sec). Leaf size: 85

from sympy import * 
t = symbols("t") 
y1 = Function("y1") 
y2 = Function("y2") 
ode=[Eq(-3*y1(t)*sec(t) - 5*y2(t)*sec(t) + Derivative(y1(t), t),0),Eq(y1(t)*sec(t) + 3*y2(t)*sec(t) + Derivative(y2(t), t),0)] 
ics = {y1(0): 2, y2(0): 1} 
dsolve(ode,func=[y1(t),y2(t)],ics=ics)
 

\[ \left [ y_{1}{\left (t \right )} = \frac {7 \sin {\left (t \right )}}{4 \left (\sin {\left (t \right )} + 1\right )} - \frac {7}{4 \left (\sin {\left (t \right )} + 1\right )} - \frac {15 \sin {\left (t \right )}}{4 \left (\sin {\left (t \right )} - 1\right )} - \frac {15}{4 \left (\sin {\left (t \right )} - 1\right )}, \ y_{2}{\left (t \right )} = - \frac {7 \sin {\left (t \right )}}{4 \left (\sin {\left (t \right )} + 1\right )} + \frac {7}{4 \left (\sin {\left (t \right )} + 1\right )} + \frac {3 \sin {\left (t \right )}}{4 \left (\sin {\left (t \right )} - 1\right )} + \frac {3}{4 \left (\sin {\left (t \right )} - 1\right )}\right ] \]