15.17.5 problem 5

Internal problem ID [3241]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 26, page 115
Problem number : 5
Date solved : Monday, January 27, 2025 at 07:27:19 AM
CAS classification : system_of_ODEs

\begin{align*} 2 \frac {d}{d t}x \left (t \right )+3 x \left (t \right )-y \left (t \right )&={\mathrm e}^{t}\\ 5 x \left (t \right )-3 \frac {d}{d t}y \left (t \right )&=y \left (t \right )+2 t \end{align*}

✓ Solution by Maple

Time used: 0.048 (sec). Leaf size: 49

dsolve([2*diff(x(t),t)+3*x(t)-y(t)=exp(t),5*x(t)-3*diff(y(t),t)=y(t)+2*t],singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {3 \,{\mathrm e}^{\frac {t}{6}} c_2}{10}-c_{1} {\mathrm e}^{-2 t}+\frac {11}{2}+\frac {4 \,{\mathrm e}^{t}}{15}+t \\ y \left (t \right ) &= {\mathrm e}^{\frac {t}{6}} c_2 +c_{1} {\mathrm e}^{-2 t}+3 t +\frac {37}{2}+\frac {{\mathrm e}^{t}}{3} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.788 (sec). Leaf size: 105

DSolve[{2*D[x[t],t]+3*x[t]-y[t]==Exp[t],5*x[t]-3*D[y[t],t]==y[t]+2*t},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to t+\frac {4 e^t}{15}+\frac {1}{13} (10 c_1-3 c_2) e^{-2 t}+\frac {3}{13} (c_1+c_2) e^{t/6}+\frac {11}{2} \\ y(t)\to \frac {1}{78} e^{-2 t} \left (39 e^{2 t} (6 t+37)+26 e^{3 t}+60 (c_1+c_2) e^{13 t/6}-60 c_1+18 c_2\right ) \\ \end{align*}