83.13.7 problem 12

Internal problem ID [22015]
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 : 12
Date solved : Sunday, October 12, 2025 at 05:52:18 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.058 (sec). Leaf size: 57

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

\begin{align*} x \left (t \right ) &= -\frac {t^{2} {\mathrm e}^{t}}{4}+c_1 \,{\mathrm e}^{t}+c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{t} t \\ y \left (t \right ) &= 2 c_2 \,{\mathrm e}^{-t}+c_3 \,{\mathrm e}^{t}-\frac {t \,{\mathrm e}^{t}}{2}+\frac {{\mathrm e}^{t}}{2}-{\mathrm e}^{-t} \\ \end{align*}

✓ Mathematica. Time used: 0.206 (sec). Leaf size: 114

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

\begin{align*} x(t)&\to \frac {1}{8} e^t \left (-2 t^2+(-2-4 c_1+4 c_2+4 c_3) t+1+6 c_1+2 c_2-2 c_3\right )+\frac {1}{4} (1+c_1-c_2+c_3) e^{-t}\\ y(t)&\to \frac {1}{2} (-1+c_1-c_2+c_3) e^{-t}+\frac {1}{4} e^t (-2 t+1-2 c_1+2 c_2+2 c_3) \end{align*}

✓ Sympy. Time used: 0.173 (sec). Leaf size: 66

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

\[ \left [ x{\left (t \right )} = - \frac {t^{2} e^{t}}{4} + t \left (C_{1} - \frac {1}{4}\right ) e^{t} + \left (\frac {C_{2}}{2} + \frac {1}{4}\right ) e^{- t} + \left (- C_{1} + C_{3} + \frac {1}{8}\right ) e^{t}, \ y{\left (t \right )} = - \frac {t e^{t}}{2} + \left (C_{1} + \frac {1}{4}\right ) e^{t} + \left (C_{2} - \frac {1}{2}\right ) e^{- t}\right ] \]