49.24.3 problem 5

Internal problem ID [7771]
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 : 5
Date solved : Wednesday, March 05, 2025 at 05:03:19 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 )&=y_{1} \left (x \right )+y_{2} \left (x \right )+{\mathrm e}^{3 x} \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.039 (sec). Leaf size: 35

ode:=[diff(y__1(x),x) = y__1(x)+y__2(x), diff(y__2(x),x) = y__1(x)+y__2(x)+exp(3*x)]; 
ic:=y__1(0) = 0y__2(0) = 0; 
dsolve([ode,ic]);
 

\begin{align*} y_{1} \left (x \right ) &= -\frac {{\mathrm e}^{2 x}}{2}+\frac {{\mathrm e}^{3 x}}{3}+\frac {1}{6} \\ y_{2} \left (x \right ) &= -\frac {{\mathrm e}^{2 x}}{2}+\frac {2 \,{\mathrm e}^{3 x}}{3}-\frac {1}{6} \\ \end{align*}

✓ Mathematica. Time used: 0.086 (sec). Leaf size: 46

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

\begin{align*} \text {y1}(x)\to \frac {1}{6} \left (e^x-1\right )^2 \left (2 e^x+1\right ) \\ \text {y2}(x)\to \frac {1}{6} \left (-3 e^{2 x}+4 e^{3 x}-1\right ) \\ \end{align*}

✓ Sympy. Time used: 0.118 (sec). Leaf size: 36

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(-y__1(x) - y__2(x) - exp(3*x) + Derivative(y__2(x), x),0)] 
ics = {} 
dsolve(ode,func=[y__1(x),y__2(x)],ics=ics)
 

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