71.18.18 problem 16

Internal problem ID [14513]
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 : 16
Date solved : Thursday, March 13, 2025 at 03:32:00 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.059 (sec). Leaf size: 70

ode:=[diff(y__1(x),x) = y__2(x)+y__4(x), diff(y__2(x),x) = y__1(x)-y__3(x), diff(y__3(x),x) = y__4(x), diff(y__4(x),x) = y__3(x)]; 
dsolve(ode);
 

\begin{align*} y_{1} \left (x \right ) &= c_{1} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{2} \\ y_{2} \left (x \right ) &= -c_{1} {\mathrm e}^{-x}+{\mathrm e}^{x} c_{2} +c_{3} {\mathrm e}^{-x}-c_4 \,{\mathrm e}^{x} \\ y_{3} \left (x \right ) &= c_{3} {\mathrm e}^{-x}+c_4 \,{\mathrm e}^{x} \\ y_{4} \left (x \right ) &= -c_{3} {\mathrm e}^{-x}+c_4 \,{\mathrm e}^{x} \\ \end{align*}

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 148

ode={D[ y1[x],x]==0*y1[x]+1*y2[x]+0*y3[x]+1*y4[x],D[ y2[x],x]==1*y1[x]+0*y2[x]-1*y3[x]+0*y4[x],D[ y3[x],x]==0*y1[x]+0*y2[x]+0*y3[x]+1*y4[x],D[ y4[x],x]==0*y1[x]+0*y2[x]+1*y3[x]+0*y4[x]}; 
ic={}; 
DSolve[{ode,ic},{y1[x],y2[x],y3[x],y4[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(x)\to \frac {1}{2} e^{-x} \left (c_1 \left (e^{2 x}+1\right )+(c_2+c_4) \left (e^{2 x}-1\right )\right ) \\ \text {y2}(x)\to \frac {1}{2} e^{-x} \left (c_1 \left (e^{2 x}-1\right )+c_2 e^{2 x}-c_3 e^{2 x}+c_2+c_3\right ) \\ \text {y3}(x)\to \frac {1}{2} e^{-x} \left (c_3 \left (e^{2 x}+1\right )+c_4 \left (e^{2 x}-1\right )\right ) \\ \text {y4}(x)\to \frac {1}{2} e^{-x} \left (c_3 \left (e^{2 x}-1\right )+c_4 \left (e^{2 x}+1\right )\right ) \\ \end{align*}

✓ Sympy. Time used: 0.155 (sec). Leaf size: 51

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
y__3 = Function("y__3") 
y__4 = Function("y__4") 
ode=[Eq(-y__2(x) - y__4(x) + Derivative(y__1(x), x),0),Eq(-y__1(x) + y__3(x) + Derivative(y__2(x), x),0),Eq(-y__4(x) + Derivative(y__3(x), x),0),Eq(-y__3(x) + Derivative(y__4(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x),y__3(x),y__4(x)],ics=ics)
 

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