81.8.7 problem 14

Internal problem ID [18646]
Book : A short course on differential equations. By Donald Francis Campbell. Maxmillan company. London. 1907
Section : Chapter VII. Ordinary differential equations in two dependent variables. Exercises at page 86
Problem number : 14
Date solved : Monday, March 31, 2025 at 05:48:47 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}z \left (x \right )+\frac {d}{d x}y \left (x \right )+5 y \left (x \right )-3 z \left (x \right )&=x +{\mathrm e}^{x}\\ \frac {d}{d x}y \left (x \right )+2 y \left (x \right )-z \left (x \right )&={\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.171 (sec). Leaf size: 55

ode:=[diff(z(x),x)+diff(y(x),x)+5*y(x)-3*z(x) = x+exp(x), diff(y(x),x)+2*y(x)-z(x) = exp(x)]; 
dsolve(ode);
 

\begin{align*} y \left (x \right ) &= {\mathrm e}^{-x} c_1 +\frac {{\mathrm e}^{x} c_2}{3}-\frac {{\mathrm e}^{x} x}{2}+\frac {3 \,{\mathrm e}^{x}}{4}-x \\ z \left (x \right ) &= {\mathrm e}^{x} c_2 +{\mathrm e}^{-x} c_1 -\frac {3 \,{\mathrm e}^{x} x}{2}+\frac {3 \,{\mathrm e}^{x}}{4}-2 x -1 \\ \end{align*}

✓ Mathematica. Time used: 0.209 (sec). Leaf size: 98

ode={D[z[x],x]+D[y[x],x]+5*y[x]-3*z[x]==x+Exp[x],D[y[x],x]+2*y[x]-z[x]==Exp[x]}; 
ic={}; 
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{4} e^{-x} \left (-4 e^x x+e^{2 x} (-2 x+3-2 c_1+2 c_2)+6 c_1-2 c_2\right ) \\ z(x)\to \frac {1}{4} e^{-x} \left (-4 e^x (2 x+1)+e^{2 x} (-6 x+3-6 c_1+6 c_2)+6 c_1-2 c_2\right ) \\ \end{align*}

✓ Sympy. Time used: 0.190 (sec). Leaf size: 54

from sympy import * 
x = symbols("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-x + 5*y(x) - 3*z(x) - exp(x) + Derivative(y(x), x) + Derivative(z(x), x),0),Eq(2*y(x) - z(x) - exp(x) + Derivative(y(x), x),0)] 
ics = {} 
dsolve(ode,func=[y(x),z(x)],ics=ics)
 

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