65.18.14 problem 12

Internal problem ID [15875]
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 : 12
Date solved : Thursday, October 02, 2025 at 10:29:00 AM
CAS classification : system_of_ODEs

\begin{align*} y_{1}^{\prime }\left (x \right )&=y_{1} \left (x \right )+y_{2} \left (x \right )+2 y_{3} \left (x \right )\\ y_{2}^{\prime }\left (x \right )&=y_{1} \left (x \right )+y_{2} \left (x \right )+2 y_{3} \left (x \right )\\ y_{3}^{\prime }\left (x \right )&=2 y_{1} \left (x \right )+2 y_{2} \left (x \right )+4 y_{3} \left (x \right ) \end{align*}

✓ Maple. Time used: 0.142 (sec). Leaf size: 41

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 114

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

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

✓ Sympy. Time used: 0.061 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
ode=[Eq(-y__1(x) - y__2(x) - 2*y__3(x) + Derivative(y__1(x), x),0),Eq(-y__1(x) - y__2(x) - 2*y__3(x) + Derivative(y__2(x), x),0),Eq(-2*y__1(x) - 2*y__2(x) - 4*y__3(x) + Derivative(y__3(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x),y__3(x)],ics=ics)
 

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