81.8.6 problem 13

Internal problem ID [18645]
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 : 13
Date solved : Monday, March 31, 2025 at 05:48:46 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y \left (x \right )+\frac {d}{d x}z \left (x \right )+6 y \left (x \right )&=0\\ \frac {d}{d x}z \left (x \right )+5 y \left (x \right )+z \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.131 (sec). Leaf size: 55

ode:=[diff(y(x),x)+diff(z(x),x)+6*y(x) = 0, diff(z(x),x)+5*y(x)+z(x) = 0]; 
dsolve(ode);
 

\begin{align*} y \left (x \right ) &= {\mathrm e}^{-x} \left (c_2 \cos \left (\sqrt {5}\, x \right )+c_1 \sin \left (\sqrt {5}\, x \right )\right ) \\ z \left (x \right ) &= {\mathrm e}^{-x} \sqrt {5}\, \left (\cos \left (\sqrt {5}\, x \right ) c_1 -\sin \left (\sqrt {5}\, x \right ) c_2 \right ) \\ \end{align*}

✓ Mathematica. Time used: 0.012 (sec). Leaf size: 81

ode={D[y[x],x]+D[z[x],x]+6*y[x]==0,D[z[x],x]+5*y[x]+z[x]==0}; 
ic={}; 
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{5} e^{-x} \left (5 c_1 \cos \left (\sqrt {5} x\right )+\sqrt {5} c_2 \sin \left (\sqrt {5} x\right )\right ) \\ z(x)\to e^{-x} \left (c_2 \cos \left (\sqrt {5} x\right )-\sqrt {5} c_1 \sin \left (\sqrt {5} x\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.164 (sec). Leaf size: 71

from sympy import * 
x = symbols("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(6*y(x) + Derivative(y(x), x) + Derivative(z(x), x),0),Eq(5*y(x) + z(x) + Derivative(z(x), x),0)] 
ics = {} 
dsolve(ode,func=[y(x),z(x)],ics=ics)
 

\[ \left [ y{\left (x \right )} = \frac {\sqrt {5} C_{1} e^{- x} \sin {\left (\sqrt {5} x \right )}}{5} + \frac {\sqrt {5} C_{2} e^{- x} \cos {\left (\sqrt {5} x \right )}}{5}, \ z{\left (x \right )} = C_{1} e^{- x} \cos {\left (\sqrt {5} x \right )} - C_{2} e^{- x} \sin {\left (\sqrt {5} x \right )}\right ] \]