81.8.1 problem 8

Internal problem ID [18640]
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 : 8
Date solved : Monday, March 31, 2025 at 05:48:39 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}z \left (x \right )+7 y \left (x \right )-3 z \left (x \right )&=0\\ 7 \frac {d}{d x}y \left (x \right )+63 y \left (x \right )-36 z \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.107 (sec). Leaf size: 32

ode:=[diff(z(x),x)+7*y(x)-3*z(x) = 0, 7*diff(y(x),x)+63*y(x)-36*z(x) = 0]; 
dsolve(ode);
 

\begin{align*} y \left (x \right ) &= {\mathrm e}^{-3 x} \left (c_2 x +c_1 \right ) \\ z \left (x \right ) &= \frac {7 \,{\mathrm e}^{-3 x} \left (6 c_2 x +6 c_1 +c_2 \right )}{36} \\ \end{align*}

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 50

ode={D[z[x],x]+7*y[x]-3*z[x]==0,7*D[y[x],x]+63*y[x]-36*z[x]==0}; 
ic={}; 
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)\to \frac {1}{7} e^{-3 x} (c_1 (7-42 x)+36 c_2 x) \\ z(x)\to e^{-3 x} (-7 c_1 x+6 c_2 x+c_2) \\ \end{align*}

✓ Sympy. Time used: 0.098 (sec). Leaf size: 46

from sympy import * 
x = symbols("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(7*y(x) - 3*z(x) + Derivative(z(x), x),0),Eq(63*y(x) - 36*z(x) + 7*Derivative(y(x), x),0)] 
ics = {} 
dsolve(ode,func=[y(x),z(x)],ics=ics)
 

\[ \left [ y{\left (x \right )} = - 6 C_{2} x e^{- 3 x} - \left (6 C_{1} - C_{2}\right ) e^{- 3 x}, \ z{\left (x \right )} = - 7 C_{1} e^{- 3 x} - 7 C_{2} x e^{- 3 x}\right ] \]