23.4.8 problem 9

Internal problem ID [4173]
Book : Theory and solutions of Ordinary Differential equations, Donald Greenspan, 1960
Section : Chapter 6. Linear systems. Exercises at page 110
Problem number : 9
Date solved : Sunday, March 30, 2025 at 02:41:35 AM
CAS classification : system_of_ODEs

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

With initial conditions

\begin{align*} y_{1} \left (0\right ) = 3\\ y_{2} \left (0\right ) = 7 \end{align*}

✓ Maple. Time used: 0.121 (sec). Leaf size: 25

ode:=[2*diff(y__1(x),x) = y__1(x)+y__2(x), 2*diff(y__2(x),x) = 5*y__2(x)-3*y__1(x)]; 
ic:=y__1(0) = 3y__2(0) = 7; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (x \right ) &= {\mathrm e}^{x}+2 \,{\mathrm e}^{2 x} \\ y_{2} \left (x \right ) &= {\mathrm e}^{x}+6 \,{\mathrm e}^{2 x} \\ \end{align*}

✓ Mathematica. Time used: 0.004 (sec). Leaf size: 30

ode={2*D[y1[x],x]==y1[x]+y2[x],2*D[y2[x],x]==5*y2[x]-3*y1[x]}; 
ic={y1[0]==3,y2[0]==7}; 
DSolve[{ode,ic},{y1[x],y2[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} \text {y1}(x)\to e^x \left (2 e^x+1\right ) \\ \text {y2}(x)\to e^x \left (6 e^x+1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.076 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y__1 = Function("y__1") 
y__2 = Function("y__2") 
ode=[Eq(-y__1(x) - y__2(x) + 2*Derivative(y__1(x), x),0),Eq(3*y__1(x) - 5*y__2(x) + 2*Derivative(y__2(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x)],ics=ics)
 

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