83.13.3 problem A(3)

Internal problem ID [22011]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter IX. System of equations. Ex. XVII at page 154
Problem number : A(3)
Date solved : Sunday, October 12, 2025 at 05:52:18 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.056 (sec). Leaf size: 29

ode:=[diff(diff(x(t),t),t)-x(t)+y(t) = exp(t), diff(x(t),t)+x(t)-diff(y(t),t)-y(t) = 3*exp(t)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.239 (sec). Leaf size: 93

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

\begin{align*} x(t)&\to 4 e^t+2 (c_1-c_2) e^{-t}+\left (-\frac {8}{3}+c_1-c_2+c_3\right ) t-c_1+2 c_2\\ y(t)&\to \frac {5 e^t}{2}+(c_1-c_2) e^{-t}+\left (-\frac {8}{3}+c_1-c_2+c_3\right ) t-c_1+2 c_2 \end{align*}

✓ Sympy. Time used: 0.121 (sec). Leaf size: 37

from sympy import * 
t = symbols("t") 
x = Function("x") 
y = Function("y") 
ode=[Eq(-x(t) + y(t) - exp(t) + Derivative(y(t), (t, 2)),0),Eq(x(t) - y(t) - 3*exp(t) + Derivative(x(t), t) - Derivative(y(t), t),0)] 
ics = {} 
dsolve(ode,func=[x(t),y(t)],ics=ics)
 

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