23.4.9 problem 10

Internal problem ID [4174]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 6. Linear systems. Exercises at page 110
Problem number : 10
Date solved : Tuesday, March 04, 2025 at 05:54:54 PM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} y_{1} \left (0\right ) = -1\\ y_{2} \left (0\right ) = 1 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 24

ode:=[diff(y__1(x),x) = -2*y__2(x), diff(y__2(x),x) = y__1(x)+2*y__2(x)]; 
ic:=y__1(0) = -1y__2(0) = 1; 
dsolve([ode,ic]);
 

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

✓ Mathematica. Time used: 0.007 (sec). Leaf size: 24

ode={D[y1[x],x]==-2*y2[x],D[y2[x],x]==y1[x]+2*y2[x]}; 
ic={y1[0]==-1,y2[0]==1}; 
DSolve[{ode,ic},{y1[x],y2[x]},x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.083 (sec). Leaf size: 42

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

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