34.15.5 problem 17

Internal problem ID [8077]
Book : Schaums Outline. Theory and problems of Differential Equations, 1st edition. Frank Ayres. McGraw Hill 1952
Section : Chapter 21. System of simultaneous linear equations. Supplemetary problems. Page 163
Problem number : 17
Date solved : Tuesday, September 30, 2025 at 05:15:16 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 )+2 y \left (t \right )&=1+{\mathrm e}^{t}\\ \frac {d}{d t}y \left (t \right )+2 y \left (t \right )+\frac {d}{d t}z \left (t \right )+z \left (t \right )&=2+{\mathrm e}^{t}\\ \frac {d}{d t}x \left (t \right )-x \left (t \right )+\frac {d}{d t}z \left (t \right )+z \left (t \right )&=3+{\mathrm e}^{t} \end{align*}

✓ Maple. Time used: 0.185 (sec). Leaf size: 44

ode:=[diff(x(t),t)-x(t)+diff(y(t),t)+2*y(t) = 1+exp(t), diff(y(t),t)+2*y(t)+diff(z(t),t)+z(t) = exp(t)+2, diff(x(t),t)-x(t)+diff(z(t),t)+z(t) = 3+exp(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= -1+\frac {{\mathrm e}^{t} \left (2 c_3 +t \right )}{2} \\ y \left (t \right ) &= \frac {{\mathrm e}^{t}}{6}+{\mathrm e}^{-2 t} c_2 \\ z \left (t \right ) &= 2+\frac {{\mathrm e}^{t}}{4}+{\mathrm e}^{-t} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.044 (sec). Leaf size: 60

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

\begin{align*} x(t)&\to -1+e^t \left (\frac {t}{2}+c_1\right )\\ y(t)&\to \frac {e^t}{6}+c_2 e^{-2 t}\\ z(t)&\to \frac {e^t}{4}+(4+c_3) e^{-t}+2 \end{align*}

✓ Sympy. Time used: 0.119 (sec). Leaf size: 42

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-x(t) + 2*y(t) - exp(t) + Derivative(x(t), t) + Derivative(y(t), t) - 1,0),Eq(2*y(t) + z(t) - exp(t) + Derivative(y(t), t) + Derivative(z(t), t) - 2,0),Eq(-x(t) + z(t) - exp(t) + Derivative(x(t), t) + Derivative(z(t), t) - 3,0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t),z(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = C_{1} e^{t} + \frac {t e^{t}}{2} - 1, \ y{\left (t \right )} = C_{2} e^{- 2 t} + \frac {e^{t}}{6}, \ z{\left (t \right )} = C_{3} e^{- t} + \frac {e^{t}}{4} + 2\right ] \]