71.18.9 problem 7

Internal problem ID [14504]
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 : 7
Date solved : Thursday, March 13, 2025 at 03:31:46 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 )-2 y_{3} \left (x \right )\\ \frac {d}{d x}y_{2} \left (x \right )&=3 y_{2} \left (x \right )-2 y_{3} \left (x \right )\\ \frac {d}{d x}y_{3} \left (x \right )&=3 y_{1} \left (x \right )+y_{2} \left (x \right )-3 y_{3} \left (x \right ) \end{align*}

✓ Maple. Time used: 0.073 (sec). Leaf size: 64

ode:=[diff(y__1(x),x) = 2*y__1(x)+y__2(x)-2*y__3(x), diff(y__2(x),x) = 3*y__2(x)-2*y__3(x), diff(y__3(x),x) = 3*y__1(x)+y__2(x)-3*y__3(x)]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 159

ode={D[ y1[x],x]==2*y1[x]+y2[x]-2*y3[x],D[ y2[x],x]==3*y2[x]-2*y3[x],D[ y3[x],x]==3*y1[x]+y2[x]-3*y3[x]}; 
ic={}; 
DSolve[{ode,ic},{y1[x],y2[x],y3[x]},x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.137 (sec). Leaf size: 61

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) + 2*y__3(x) + Derivative(y__1(x), x),0),Eq(-3*y__2(x) + 2*y__3(x) + Derivative(y__2(x), x),0),Eq(-3*y__1(x) - y__2(x) + 3*y__3(x) + Derivative(y__3(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x),y__3(x)],ics=ics)
 

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