71.18.13 problem 11

Internal problem ID [14508]
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 : 11
Date solved : Thursday, March 13, 2025 at 03:31:51 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.080 (sec). Leaf size: 50

ode:=[diff(y__1(x),x) = -2*y__1(x)-y__2(x)+y__3(x), diff(y__2(x),x) = -y__1(x)-2*y__2(x)-y__3(x), diff(y__3(x),x) = y__1(x)-y__2(x)-2*y__3(x)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 130

ode={D[ y1[x],x]==-2*y1[x]-1*y2[x]+1*y3[x],D[ y2[x],x]==-1*y1[x]-2*y2[x]-1*y3[x],D[ y3[x],x]==1*y1[x]-1*y2[x]-2*y3[x]}; 
ic={}; 
DSolve[{ode,ic},{y1[x],y2[x],y3[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(x)\to \frac {1}{3} e^{-3 x} \left (c_1 \left (e^{3 x}+2\right )-(c_2-c_3) \left (e^{3 x}-1\right )\right ) \\ \text {y2}(x)\to \frac {1}{3} e^{-3 x} \left (-\left (c_1 \left (e^{3 x}-1\right )\right )+c_2 \left (e^{3 x}+2\right )-c_3 \left (e^{3 x}-1\right )\right ) \\ \text {y3}(x)\to \frac {1}{3} e^{-3 x} \left (c_1 \left (e^{3 x}-1\right )-c_2 \left (e^{3 x}-1\right )+c_3 \left (e^{3 x}+2\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.117 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(2*y__1(x) + y__2(x) - y__3(x) + Derivative(y__1(x), x),0),Eq(y__1(x) + 2*y__2(x) + y__3(x) + Derivative(y__2(x), x),0),Eq(-y__1(x) + y__2(x) + 2*y__3(x) + Derivative(y__3(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x),y__3(x)],ics=ics)
 

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