49.24.1 problem 3

Internal problem ID [7769]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 6. Existence and uniqueness of solutions to systems and nth order equations. Page 250
Problem number : 3
Date solved : Wednesday, March 05, 2025 at 05:03:16 AM
CAS classification : system_of_ODEs

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

With initial conditions

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

✓ Maple. Time used: 0.036 (sec). Leaf size: 15

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 18

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

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

✓ Sympy. Time used: 0.067 (sec). Leaf size: 20

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

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