40.15.5 problem 17

Internal problem ID [6791]
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 : Monday, January 27, 2025 at 02:30:36 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 )&={\mathrm e}^{t}+2\\ \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*}

✓ Solution by Maple

Time used: 0.033 (sec). Leaf size: 44

dsolve([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)=2+exp(t),diff(x(t),t)-x(t)+diff(z(t),t)+z(t)=3+exp(t)],singsol=all)
 

\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*}

✓ Solution by Mathematica

Time used: 0.076 (sec). Leaf size: 60

DSolve[{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]},{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*}