38.1.48 problem 50

Internal problem ID [8209]
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 : Sunday, October 12, 2025 at 01:34:59 AM
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*}

✓ Maple. Time used: 0.209 (sec). Leaf size: 67

ode:=[diff(diff(x(t),t),t) = 4*y(t)+exp(t), diff(diff(y(t),t),t) = 4*x(t)-exp(t)]; 
dsolve(ode);
 

\begin{align*} x \left (t \right ) &= \frac {{\mathrm e}^{t}}{5}+c_1 \cos \left (2 t \right )+c_2 \,{\mathrm e}^{-2 t}+c_3 \,{\mathrm e}^{2 t}+c_4 \sin \left (2 t \right ) \\ y \left (t \right ) &= -\frac {{\mathrm e}^{t}}{5}-c_1 \cos \left (2 t \right )+c_2 \,{\mathrm e}^{-2 t}+c_3 \,{\mathrm e}^{2 t}-c_4 \sin \left (2 t \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.045 (sec). Leaf size: 500

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

\begin{align*} x(t)&\to \frac {1}{8} e^{-2 t} \left (2 \left (e^{4 t}-2 e^{2 t} \cos (2 t)+1\right ) \int _1^te^{K[3]} \cos (K[3]) \sin (K[3])dK[3]+\left (e^{4 t}+2 e^{2 t} \sin (2 t)-1\right ) \int _1^te^{K[2]} \cos (2 K[2])dK[2]+\left (e^{4 t}-2 e^{2 t} \sin (2 t)-1\right ) \int _1^t-e^{K[4]} \cos (2 K[4])dK[4]+2 \left (e^{4 t}+2 e^{2 t} \cos (2 t)+1\right ) \int _1^t-e^{K[1]} \cos (K[1]) \sin (K[1])dK[1]+2 c_3 \left (e^{4 t}-2 e^{2 t} \cos (2 t)+1\right )+2 c_1 \left (e^{4 t}+2 e^{2 t} \cos (2 t)+1\right )+c_4 \left (e^{4 t}-2 e^{2 t} \sin (2 t)-1\right )+c_2 \left (e^{4 t}+2 e^{2 t} \sin (2 t)-1\right )\right )\\ y(t)&\to \frac {1}{8} e^{-2 t} \left (2 \left (e^{4 t}-2 e^{2 t} \cos (2 t)+1\right ) \int _1^t-e^{K[1]} \cos (K[1]) \sin (K[1])dK[1]+\left (e^{4 t}-2 e^{2 t} \sin (2 t)-1\right ) \int _1^te^{K[2]} \cos (2 K[2])dK[2]+\left (e^{4 t}+2 e^{2 t} \sin (2 t)-1\right ) \int _1^t-e^{K[4]} \cos (2 K[4])dK[4]+2 \left (e^{4 t}+2 e^{2 t} \cos (2 t)+1\right ) \int _1^te^{K[3]} \cos (K[3]) \sin (K[3])dK[3]+2 c_1 \left (e^{4 t}-2 e^{2 t} \cos (2 t)+1\right )+2 c_3 \left (e^{4 t}+2 e^{2 t} \cos (2 t)+1\right )+c_2 \left (e^{4 t}-2 e^{2 t} \sin (2 t)-1\right )+c_4 \left (e^{4 t}+2 e^{2 t} \sin (2 t)-1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.190 (sec). Leaf size: 119

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-4*y(t) - exp(t) + Derivative(x(t), (t, 2)),0),Eq(-4*x(t) + exp(t) + Derivative(y(t), (t, 2)),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

\[ \left [ x{\left (t \right )} = - \frac {C_{1} e^{- 2 t}}{2} + \frac {C_{2} e^{2 t}}{2} - \frac {C_{3} \sin {\left (2 t \right )}}{2} - \frac {C_{4} \cos {\left (2 t \right )}}{2} + \frac {e^{t} \sin ^{2}{\left (2 t \right )}}{5} + \frac {e^{t} \cos ^{2}{\left (2 t \right )}}{5}, \ y{\left (t \right )} = - \frac {C_{1} e^{- 2 t}}{2} + \frac {C_{2} e^{2 t}}{2} + \frac {C_{3} \sin {\left (2 t \right )}}{2} + \frac {C_{4} \cos {\left (2 t \right )}}{2} - \frac {e^{t} \sin ^{2}{\left (2 t \right )}}{5} - \frac {e^{t} \cos ^{2}{\left (2 t \right )}}{5}\right ] \]