44.1.48 problem 50

Internal problem ID [6923]
Book : A First Course in Differential Equations with Modeling Applications by Dennis G. Zill. 12 ed. Metric version. 2024. Cengage learning.
Section : Chapter 1. Introduction to differential equations. Exercises 1.1 at page 12
Problem number : 50
Date solved : Tuesday, January 28, 2025 at 03:10:36 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d^{2}}{d t^{2}}x \left (t \right )&=4 y \left (t \right )+{\mathrm e}^{t}\\ \frac {d^{2}}{d t^{2}}y \left (t \right )&=4 x \left (t \right )-{\mathrm e}^{t} \end{align*}

✓ Solution by Maple

Time used: 0.035 (sec). Leaf size: 67

dsolve([diff(x(t),t$2)=4*y(t)+exp(t),diff(y(t),t$2)=4*x(t)-exp(t)],singsol=all)
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{t}}{5}+\cos \left (2 t \right ) c_{1} +{\mathrm e}^{-2 t} c_{2} +c_3 \,{\mathrm e}^{2 t}+c_4 \sin \left (2 t \right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{t}}{5}-\cos \left (2 t \right ) c_{1} +{\mathrm e}^{-2 t} c_{2} +c_3 \,{\mathrm e}^{2 t}-c_4 \sin \left (2 t \right ) \\ \end{align*}

✓ Solution by Mathematica

Time used: 0.067 (sec). Leaf size: 218

DSolve[{D[x[t],{t,2}]==4*y[t]+Exp[t],D[y[t],{t,2}]==4*x[t]-Exp[t]},{x[t],y[t]},t,IncludeSingularSolutions -> True]
 

\begin{align*} x(t)\to \frac {1}{40} e^{-2 t} \left (8 e^{3 t}+10 c_1 e^{4 t}+5 c_2 e^{4 t}+10 c_3 e^{4 t}+5 c_4 e^{4 t}+20 (c_1-c_3) e^{2 t} \cos (2 t)+10 (c_2-c_4) e^{2 t} \sin (2 t)+10 c_1-5 c_2+10 c_3-5 c_4\right ) \\ y(t)\to \frac {1}{40} e^{-2 t} \left (-8 e^{3 t}+10 c_1 e^{4 t}+5 c_2 e^{4 t}+10 c_3 e^{4 t}+5 c_4 e^{4 t}-20 (c_1-c_3) e^{2 t} \cos (2 t)-10 (c_2-c_4) e^{2 t} \sin (2 t)+10 c_1-5 c_2+10 c_3-5 c_4\right ) \\ \end{align*}