22.4.6 problem 7.6

Internal problem ID [4538]
Book : Differential equations for engineers by Wei-Chau XIE, Cambridge Press 2010
Section : Chapter 7. Systems of linear differential equations. Problems at page 351
Problem number : 7.6
Date solved : Tuesday, September 30, 2025 at 07:34:06 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y_{1} \left (x \right )-y_{2} \left (x \right )&=0\\ 4 y_{1} \left (x \right )+\frac {d}{d x}y_{2} \left (x \right )-4 y_{2} \left (x \right )-2 y_{3} \left (x \right )&=0\\ -2 y_{1} \left (x \right )+y_{2} \left (x \right )+\frac {d}{d x}y_{3} \left (x \right )+y_{3} \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.121 (sec). Leaf size: 43

ode:=[diff(y__1(x),x)-y__2(x) = 0, 4*y__1(x)+diff(y__2(x),x)-4*y__2(x)-2*y__3(x) = 0, -2*y__1(x)+y__2(x)+diff(y__3(x),x)+y__3(x) = 0]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 119

ode={D[y1[x],x]-y2[x]==0,4*y1[x]+D[y2[x],x]-4*y2[x]-2*y3[x]==0, -2*y1[x]+y2[x]+D[y3[x],x]+y3[x]==0}; 
ic={}; 
DSolve[{ode,ic},{y1[x],y2[x],y3[x]},x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.065 (sec). Leaf size: 44

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

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