88.23.18 problem 20

Internal problem ID [24192]
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 : 20
Date solved : Thursday, October 02, 2025 at 10:00:39 PM
CAS classification : system_of_ODEs

\begin{align*} x \left (t \right )-y \left (t \right )+\frac {d}{d t}z \left (t \right )&=0\\ \frac {d}{d t}x \left (t \right )-y \left (t \right )&=1\\ \frac {d}{d t}y \left (t \right )-y \left (t \right )+z \left (t \right )&=0 \end{align*}

✓ Maple. Time used: 0.067 (sec). Leaf size: 58

ode:=[x(t)-y(t)+diff(z(t),t) = 0, diff(x(t),t)-y(t) = 1, diff(y(t),t)-y(t)+z(t) = 0]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= -1+c_2 \,{\mathrm e}^{t}+c_1 \sin \left (t \right )-c_3 \cos \left (t \right ) \\ y \left (t \right ) &= -1+c_1 \cos \left (t \right )+c_2 \,{\mathrm e}^{t}+c_3 \sin \left (t \right ) \\ z \left (t \right ) &= c_1 \sin \left (t \right )-c_3 \cos \left (t \right )-1+c_1 \cos \left (t \right )+c_3 \sin \left (t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.038 (sec). Leaf size: 127

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

\begin{align*} x(t)&\to \frac {1}{2} \left (c_1 e^t+c_2 e^t-c_3 e^t+(c_1-c_2+c_3) \cos (t)+(-c_1+c_2+c_3) \sin (t)-2\right )\\ y(t)&\to \frac {1}{2} \left (c_1 e^t+c_2 e^t-c_3 e^t+(-c_1+c_2+c_3) \cos (t)-(c_1-c_2+c_3) \sin (t)-2\right )\\ z(t)&\to c_3 \cos (t)+(c_2-c_1) \sin (t)-1 \end{align*}

✓ Sympy. Time used: 0.133 (sec). Leaf size: 110

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

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