81.8.2 problem 9

Internal problem ID [18641]
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 : 9
Date solved : Monday, March 31, 2025 at 05:48:40 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}z \left (x \right )+2 \frac {d}{d x}y \left (x \right )+3 y \left (x \right )&=0\\ \frac {d}{d x}y \left (x \right )+3 y \left (x \right )-2 z \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.112 (sec). Leaf size: 34

ode:=[diff(z(x),x)+2*diff(y(x),x)+3*y(x) = 0, diff(y(x),x)+3*y(x)-2*z(x) = 0]; 
dsolve(ode);
 

\begin{align*} y \left (x \right ) &= -\frac {2 c_1 \,{\mathrm e}^{-6 x}}{3}+c_2 \,{\mathrm e}^{-x} \\ z \left (x \right ) &= c_1 \,{\mathrm e}^{-6 x}+c_2 \,{\mathrm e}^{-x} \\ \end{align*}

✓ Mathematica. Time used: 0.009 (sec). Leaf size: 145

ode={D[z[x],x]+2*D[y[x],x]-3*y[x]==0,D[y[x],x]+3*y[x]-2*z[x]==0}; 
ic={}; 
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{146} e^{-\frac {1}{2} \left (7+\sqrt {73}\right ) x} \left (c_1 \left (\left (73+\sqrt {73}\right ) e^{\sqrt {73} x}+73-\sqrt {73}\right )+4 \sqrt {73} c_2 \left (e^{\sqrt {73} x}-1\right )\right ) \\ z(x)\to \frac {1}{146} e^{-\frac {1}{2} \left (7+\sqrt {73}\right ) x} \left (18 \sqrt {73} c_1 \left (e^{\sqrt {73} x}-1\right )-c_2 \left (\left (\sqrt {73}-73\right ) e^{\sqrt {73} x}-73-\sqrt {73}\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 31

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

\[ \left [ y{\left (x \right )} = - \frac {2 C_{1} e^{- 6 x}}{3} + C_{2} e^{- x}, \ z{\left (x \right )} = C_{1} e^{- 6 x} + C_{2} e^{- x}\right ] \]