71.18.5 problem 5 a

Internal problem ID [14500]
Book : Ordinary Differential Equations by Charles E. Roberts, Jr. CRC Press. 2010
Section : Chapter 8. Linear Systems of First-Order Differential Equations. Exercises 8.3 page 379
Problem number : 5 a
Date solved : Thursday, March 13, 2025 at 03:31:43 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y_{1} \left (x \right )&=2 y_{1} \left (x \right )-3 y_{2} \left (x \right )\\ \frac {d}{d x}y_{2} \left (x \right )&=y_{1} \left (x \right )-2 y_{2} \left (x \right ) \end{align*}

✓ Maple. Time used: 0.030 (sec). Leaf size: 30

ode:=[diff(y__1(x),x) = 2*y__1(x)-3*y__2(x), diff(y__2(x),x) = y__1(x)-2*y__2(x)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (x \right ) &= c_{1} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{2} \\ y_{2} \left (x \right ) &= c_{1} {\mathrm e}^{-x}+\frac {{\mathrm e}^{x} c_{2}}{3} \\ \end{align*}

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

ode={D[ y1[x],x]==-2*y1[x]-3*y2[x],D[ y2[x],x]==y1[x]-2*y2[x]}; 
ic={}; 
DSolve[{ode,ic},{y1[x],y2[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(x)\to e^{-2 x} \left (c_1 \cos \left (\sqrt {3} x\right )-\sqrt {3} c_2 \sin \left (\sqrt {3} x\right )\right ) \\ \text {y2}(x)\to \frac {1}{3} e^{-2 x} \left (3 c_2 \cos \left (\sqrt {3} x\right )+\sqrt {3} c_1 \sin \left (\sqrt {3} x\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.085 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-2*y__1(x) + 3*y__2(x) + Derivative(y__1(x), x),0),Eq(-y__1(x) + 2*y__2(x) + Derivative(y__2(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x)],ics=ics)
 

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