20.13.15 problem 21

Internal problem ID [3824]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 9, First order linear systems. Section 9.1, page 587
Problem number : 21
Date solved : Friday, March 14, 2025 at 01:27:28 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x_{1} \left (t \right )&=-\tan \left (t \right ) x_{1} \left (t \right )+3 \cos \left (t \right )^{2}\\ \frac {d}{d t}x_{2} \left (t \right )&=x_{1} \left (t \right )+\tan \left (t \right ) x_{2} \left (t \right )+2 \sin \left (t \right ) \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.246 (sec). Leaf size: 49

ode:=[diff(x__1(t),t) = -tan(t)*x__1(t)+3*cos(t)^2, diff(x__2(t),t) = x__1(t)+tan(t)*x__2(t)+2*sin(t)]; 
ic:=x__1(0) = 4x__2(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} x_{1} \left (t \right ) &= \frac {3 \sin \left (2 t \right )}{2}+4 \cos \left (t \right ) \\ x_{2} \left (t \right ) &= -\frac {-4 \sin \left (2 t \right )-8 t +\cos \left (3 t \right )+3 \cos \left (t \right )+2 \cos \left (2 t \right )-6}{4 \cos \left (t \right )} \\ \end{align*}

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 43

ode={D[x1[t],t]==-Tan[t]*x1[t]+3*Cos[t]^2,D[x2[t],t]==x1[t]+Tan[t]*x2[t]+2*Sin[t]}; 
ic={x1[0]==4,x2[0]==0}; 
DSolve[{ode,ic},{x1[t],x2[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} \text {x1}(t)\to (3 \sin (t)+4) \cos (t) \\ \text {x2}(t)\to \sec (t) \left (2 t+\sin (2 t)-2 \cos ^2\left (\frac {t}{2}\right ) \cos ^2(t)+2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.224 (sec). Leaf size: 65

from sympy import * 
t = symbols("t") 
x__1 = Function("x__1") 
x__2 = Function("x__2") 
ode=[Eq(x__1(t)*tan(t) - 3*cos(t)**2 + Derivative(x__1(t), t),0),Eq(-x__1(t) - x__2(t)*tan(t) - 2*sin(t) + Derivative(x__2(t), t),0)] 
ics = {} 
dsolve(ode,func=[x__1(t),x__2(t)],ics=ics)
 

\[ \left [ x^{1}{\left (t \right )} = C_{1} \cos {\left (t \right )} + 3 \sin {\left (t \right )} \cos {\left (t \right )}, \ x^{2}{\left (t \right )} = \frac {C_{1} t \sin ^{2}{\left (t \right )}}{2 \cos {\left (t \right )}} + \frac {C_{1} t \cos {\left (t \right )}}{2} + \frac {C_{1} \sin {\left (t \right )}}{2} + \frac {C_{2}}{\cos {\left (t \right )}} + \frac {\sin ^{2}{\left (t \right )}}{\cos {\left (t \right )}} - \cos ^{2}{\left (t \right )}\right ] \]