23.4.3 problem 8(c)

Internal problem ID [4168]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 6. Linear systems. Exercises at page 110
Problem number : 8(c)
Date solved : Sunday, March 30, 2025 at 02:41:28 AM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.135 (sec). Leaf size: 45

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

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

✓ Mathematica. Time used: 0.006 (sec). Leaf size: 51

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

\begin{align*} \text {y1}(x)\to e^{2 x} (c_1 \cos (x)-(c_1+c_2) \sin (x)) \\ \text {y2}(x)\to e^{2 x} (c_2 \cos (x)+(2 c_1+c_2) \sin (x)) \\ \end{align*}

✓ Sympy. Time used: 0.108 (sec). Leaf size: 56

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-y__1(x) + y__2(x) + Derivative(y__1(x), x),0),Eq(-2*y__1(x) - 3*y__2(x) + Derivative(y__2(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x)],ics=ics)
 

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