81.8.3 problem 10

Internal problem ID [18642]
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 : 10
Date solved : Monday, March 31, 2025 at 05:48:41 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y \left (x \right )+3 y \left (x \right )+z \left (x \right )&=0\\ \frac {d}{d x}z \left (x \right )+3 y \left (x \right )+5 z \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.103 (sec). Leaf size: 35

ode:=[diff(y(x),x)+3*y(x)+z(x) = 0, diff(z(x),x)+3*y(x)+5*z(x) = 0]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 71

ode={D[y[x],x]+3*y[x]+z[x]==0,D[z[x],x]+3*y[x]+5*z[x]==0}; 
ic={}; 
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.096 (sec). Leaf size: 32

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

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