71.18.11 problem 9

Internal problem ID [14514]
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 : 9
Date solved : Monday, March 31, 2025 at 12:29:11 PM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y_{1} \left (x \right )&=4 y_{1} \left (x \right )+6 y_{2} \left (x \right )+6 y_{3} \left (x \right )\\ \frac {d}{d x}y_{2} \left (x \right )&=y_{1} \left (x \right )+3 y_{2} \left (x \right )+2 y_{3} \left (x \right )\\ \frac {d}{d x}y_{3} \left (x \right )&=-y_{1} \left (x \right )-4 y_{2} \left (x \right )-3 y_{3} \left (x \right ) \end{align*}

✓ Maple. Time used: 0.146 (sec). Leaf size: 62

ode:=[diff(y__1(x),x) = 4*y__1(x)+6*y__2(x)+6*y__3(x), diff(y__2(x),x) = y__1(x)+3*y__2(x)+2*y__3(x), diff(y__3(x),x) = -y__1(x)-4*y__2(x)-3*y__3(x)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (x \right ) &= c_2 \,{\mathrm e}^{-x}+c_3 \,{\mathrm e}^{4 x} \\ y_{2} \left (x \right ) &= \frac {c_2 \,{\mathrm e}^{-x}}{3}+\frac {c_3 \,{\mathrm e}^{4 x}}{3}+{\mathrm e}^{x} c_1 \\ y_{3} \left (x \right ) &= -\frac {7 c_2 \,{\mathrm e}^{-x}}{6}-\frac {c_3 \,{\mathrm e}^{4 x}}{3}-{\mathrm e}^{x} c_1 \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 145

ode={D[ y1[x],x]==4*y1[x]+6*y2[x]+6*y3[x],D[ y2[x],x]==1*y1[x]+3*y2[x]+2*y3[x],D[ y3[x],x]==-1*y1[x]-4*y2[x]-3*y3[x]}; 
ic={}; 
DSolve[{ode,ic},{y1[x],y2[x],y3[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(x)\to \frac {1}{5} e^{-x} \left ((5 c_1+6 (c_2+c_3)) e^{5 x}-6 (c_2+c_3)\right ) \\ \text {y2}(x)\to \frac {1}{15} e^{-x} \left (-5 (c_1-3 c_2) e^{2 x}+(5 c_1+6 (c_2+c_3)) e^{5 x}-6 (c_2+c_3)\right ) \\ \text {y3}(x)\to \frac {1}{3} (c_1-3 c_2) e^x+\frac {7}{5} (c_2+c_3) e^{-x}-\frac {1}{15} (5 c_1+6 (c_2+c_3)) e^{4 x} \\ \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 63

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-4*y__1(x) - 6*y__2(x) - 6*y__3(x) + Derivative(y__1(x), x),0),Eq(-y__1(x) - 3*y__2(x) - 2*y__3(x) + Derivative(y__2(x), x),0),Eq(y__1(x) + 4*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)
 

\[ \left [ y^{1}{\left (x \right )} = - \frac {6 C_{1} e^{- x}}{7} - 3 C_{2} e^{4 x}, \ y^{2}{\left (x \right )} = - \frac {2 C_{1} e^{- x}}{7} - C_{2} e^{4 x} - C_{3} e^{x}, \ y^{3}{\left (x \right )} = C_{1} e^{- x} + C_{2} e^{4 x} + C_{3} e^{x}\right ] \]