88.23.8 problem 9

Internal problem ID [24182]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 5. Special Techniques for Linear Equations. Miscellaneous Exercises at page 162
Problem number : 9
Date solved : Thursday, October 02, 2025 at 10:00:33 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d t}x \left (t \right )-x \left (t \right )-\frac {d}{d t}y \left (t \right )&=0\\ \frac {d}{d t}y \left (t \right )+3 x \left (t \right )-2 y \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.045 (sec). Leaf size: 65

ode:=[diff(x(t),t)-x(t)-diff(y(t),t) = 0, 3*x(t)+diff(y(t),t)-2*y(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= c_1 \sin \left (\sqrt {2}\, t \right )+c_2 \cos \left (\sqrt {2}\, t \right ) \\ y \left (t \right ) &= \frac {c_1 \sqrt {2}\, \cos \left (\sqrt {2}\, t \right )}{2}-\frac {c_2 \sqrt {2}\, \sin \left (\sqrt {2}\, t \right )}{2}+c_1 \sin \left (\sqrt {2}\, t \right )+c_2 \cos \left (\sqrt {2}\, t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 76

ode={D[x[t],{t,1}]-x[t]-D[y[t],t]==0,3*x[t]+D[y[t],t]-2*y[t]==0}; 
ic={}; 
DSolve[{ode,ic},{x[t],y[t]},t,IncludeSingularSolutions->True]
 

\begin{align*} x(t)&\to c_1 \cos \left (\sqrt {2} t\right )+\sqrt {2} (c_2-c_1) \sin \left (\sqrt {2} t\right )\\ y(t)&\to c_2 \cos \left (\sqrt {2} t\right )+\frac {(2 c_2-3 c_1) \sin \left (\sqrt {2} t\right )}{\sqrt {2}} \end{align*}

✓ Sympy. Time used: 0.106 (sec). Leaf size: 68

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + Derivative(x(t), t) - Derivative(y(t), t),0),Eq(3*x(t) - 2*y(t) + Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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