15.17.3 problem 3

Internal problem ID [3239]
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 : 3
Date solved : Monday, January 27, 2025 at 07:27:17 AM
CAS classification : system_of_ODEs

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

✓ Solution by Maple

Time used: 0.179 (sec). Leaf size: 45

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

\begin{align*} x \left (t \right ) &= \frac {3 t}{2}-\frac {3}{4}+{\mathrm e}^{-2 t} c_2 \\ y \left (t \right ) &= -\frac {2 \,{\mathrm e}^{-2 t} c_2}{3}+\frac {4 \sin \left (2 t \right )}{17}+\frac {\cos \left (2 t \right )}{17}-\frac {3}{2}+{\mathrm e}^{-\frac {t}{2}} c_{1} \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.310 (sec). Leaf size: 74

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

\begin{align*} x(t)\to \frac {3 t}{2}+c_1 e^{-2 t}-\frac {3}{4} \\ y(t)\to \frac {4}{17} \sin (2 t)+\frac {1}{17} \cos (2 t)+\frac {1}{6} \left (-4 c_1 e^{-2 t}+(4 c_1+6 c_2) e^{-t/2}-9\right ) \\ \end{align*}