88.11.5 problem 6

Internal problem ID [24052]
Book : Elementary Differential Equations. By Lee Roy Wilcox and Herbert J. Curtis. 1961 first edition. International texbook company. Scranton, Penn. USA. CAT number 61-15976
Section : Chapter 2. Differential equations of first order. Exercise at page 54
Problem number : 6
Date solved : Thursday, October 02, 2025 at 09:55:14 PM
CAS classification : system_of_ODEs

\begin{align*} y^{\prime }+2 z \left (x \right )&=y\\ z^{\prime }\left (x \right )+4 y&=0 \end{align*}

✓ Maple. Time used: 0.040 (sec). Leaf size: 67

ode:=[diff(y(x),x)+2*z(x) = y(x), diff(z(x),x)+4*y(x) = 0]; 
dsolve(ode);
 

\begin{align*} y \left (x \right ) &= \left (-\frac {\sqrt {33}}{8}-\frac {1}{8}\right ) c_1 \,{\mathrm e}^{\frac {\left (1+\sqrt {33}\right ) x}{2}}+\left (\frac {\sqrt {33}}{8}-\frac {1}{8}\right ) c_2 \,{\mathrm e}^{-\frac {\left (-1+\sqrt {33}\right ) x}{2}} \\ z \left (x \right ) &= c_1 \,{\mathrm e}^{\frac {\left (1+\sqrt {33}\right ) x}{2}}+c_2 \,{\mathrm e}^{-\frac {\left (-1+\sqrt {33}\right ) x}{2}} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 148

ode={D[y[x],x]+2*z[x]==y[x],D[z[x],x]+4*y[x]==0}; 
ic={}; 
DSolve[{ode,ic},{y[x],z[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to \frac {1}{66} e^{\frac {1}{2} \left (x-\sqrt {33} x\right )} \left (c_1 \left (\left (33+\sqrt {33}\right ) e^{\sqrt {33} x}+33-\sqrt {33}\right )-4 \sqrt {33} c_2 \left (e^{\sqrt {33} x}-1\right )\right )\\ z(x)&\to -\frac {1}{66} e^{\frac {1}{2} \left (x-\sqrt {33} x\right )} \left (8 \sqrt {33} c_1 \left (e^{\sqrt {33} x}-1\right )+c_2 \left (\left (\sqrt {33}-33\right ) e^{\sqrt {33} x}-33-\sqrt {33}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.115 (sec). Leaf size: 76

from sympy import * 
x = symbols("x") 
y = Function("y") 
z = Function("z") 
ode=[Eq(-y(x) + 2*z(x) + Derivative(y(x), x),0),Eq(4*y(x) + Derivative(z(x), x),0)] 
ics = {} 
dsolve(ode,func=[y(x),z(x)],ics=ics)
 

\[ \left [ y{\left (x \right )} = - \frac {C_{1} \left (1 - \sqrt {33}\right ) e^{\frac {x \left (1 - \sqrt {33}\right )}{2}}}{8} - \frac {C_{2} \left (1 + \sqrt {33}\right ) e^{\frac {x \left (1 + \sqrt {33}\right )}{2}}}{8}, \ z{\left (x \right )} = C_{1} e^{\frac {x \left (1 - \sqrt {33}\right )}{2}} + C_{2} e^{\frac {x \left (1 + \sqrt {33}\right )}{2}}\right ] \]