28.4.7 problem 7.7

Internal problem ID [4539]
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 : 7.7
Date solved : Sunday, March 30, 2025 at 03:25:46 AM
CAS classification : system_of_ODEs

\begin{align*} \frac {d}{d x}y_{1} \left (x \right )-2 y_{1} \left (x \right )+3 y_{2} \left (x \right )-3 y_{3} \left (x \right )&=0\\ -4 y_{1} \left (x \right )+\frac {d}{d x}y_{2} \left (x \right )+5 y_{2} \left (x \right )-3 y_{3} \left (x \right )&=0\\ -4 y_{1} \left (x \right )+4 y_{2} \left (x \right )+\frac {d}{d x}y_{3} \left (x \right )-2 y_{3} \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.169 (sec). Leaf size: 55

ode:=[diff(y__1(x),x)-2*y__1(x)+3*y__2(x)-3*y__3(x) = 0, -4*y__1(x)+diff(y__2(x),x)+5*y__2(x)-3*y__3(x) = 0, -4*y__1(x)+4*y__2(x)+diff(y__3(x),x)-2*y__3(x) = 0]; 
dsolve(ode);
 

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

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 91

ode={D[y1[x],x]-2*y1[x]+3*y2[x]-3*y2[x]==0,-4*y1[x]+D[y2[x],x]+5*y2[x]-3*y3[x]==0, -4*y1[x]+4*y2[x]+D[y3[x],x]-2*y3[x]==0}; 
ic={}; 
DSolve[{ode,ic},{y1[x],y2[x],y3[x]},x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.137 (sec). Leaf size: 49

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) + 3*y__2(x) - 3*y__3(x) + Derivative(y__1(x), x),0),Eq(-4*y__1(x) + 5*y__2(x) - 3*y__3(x) + Derivative(y__2(x), x),0),Eq(-4*y__1(x) + 4*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)
 

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