89.27.3 problem 3

Internal problem ID [24885]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 13. Systems of equations. Exercises at page 200
Problem number : 3
Date solved : Thursday, October 02, 2025 at 10:49:04 PM
CAS classification : system_of_ODEs

\begin{align*} y^{\prime }+y-v^{\prime }\left (x \right )-v \left (x \right )&=0\\ y^{\prime }+v^{\prime }\left (x \right )-v \left (x \right )&={\mathrm e}^{x} \end{align*}

✓ Maple. Time used: 0.050 (sec). Leaf size: 38

ode:=[diff(y(x),x)+y(x)-diff(v(x),x)-v(x) = 0, diff(y(x),x)+diff(v(x),x)-v(x) = exp(x)]; 
dsolve(ode);
 

\begin{align*} v \left (x \right ) &= {\mathrm e}^{\frac {x}{2}} c_2 +{\mathrm e}^{-x} c_1 +{\mathrm e}^{x} \\ y \left (x \right ) &= {\mathrm e}^{\frac {x}{2}} c_2 -2 \,{\mathrm e}^{-x} c_1 +{\mathrm e}^{x} \\ \end{align*}

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 86

ode={D[y[x],x]+y[x]-D[v[x],x]-v[x]==0,D[y[x],x]+D[v[x],x]-v[x]==Exp[x]}; 
ic={}; 
DSolve[{ode,ic},{y[x],v[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} v(x)&\to \frac {1}{3} e^{-x} \left (3 e^{2 x}+(2 c_1+c_2) e^{3 x/2}+c_1-c_2\right )\\ y(x)&\to \frac {1}{3} e^{-x} \left (3 e^{2 x}+(2 c_1+c_2) e^{3 x/2}-2 c_1+2 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.080 (sec). Leaf size: 36

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

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