89.27.4 problem 4

Internal problem ID [24886]
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 : 4
Date solved : Thursday, October 02, 2025 at 10:49:05 PM
CAS classification : system_of_ODEs

\begin{align*} 2 v^{\prime }\left (x \right )+2 v \left (x \right )+w^{\prime }\left (x \right )-w \left (x \right )&=3 x\\ v^{\prime }\left (x \right )+v \left (x \right )+w^{\prime }\left (x \right )+w \left (x \right )&=1 \end{align*}

✓ Maple. Time used: 0.043 (sec). Leaf size: 33

ode:=[2*diff(v(x),x)+2*v(x)+diff(w(x),x)-w(x) = 3*x, diff(v(x),x)+v(x)+diff(w(x),x)+w(x) = 1]; 
dsolve(ode);
 

\begin{align*} v \left (x \right ) &= {\mathrm e}^{-3 x} c_2 +{\mathrm e}^{-x} c_1 +x \\ w \left (x \right ) &= -{\mathrm e}^{-3 x} c_2 +1-x \\ \end{align*}

✓ Mathematica. Time used: 0.032 (sec). Leaf size: 43

ode={2*D[v[x],x]+2*v[x]+D[w[x],x]-w[x]==3*x,D[v[x],x]+v[x]+D[w[x],x]+w[x]==1}; 
ic={}; 
DSolve[{ode,ic},{w[x],v[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} v(x)&\to x-c_2 e^{-3 x}+(c_1+c_2) e^{-x}\\ w(x)&\to -x+c_2 e^{-3 x}+1 \end{align*}

✓ Sympy. Time used: 0.098 (sec). Leaf size: 27

from sympy import * 
x = symbols("x") 
w = Function("w") 
v = Function("v") 
ode=[Eq(-3*x + 2*v(x) - w(x) + 2*Derivative(v(x), x) + Derivative(w(x), x),0),Eq(v(x) + w(x) + Derivative(v(x), x) + Derivative(w(x), x) - 1,0)] 
ics = {} 
dsolve(ode,func=[w(x),v(x)],ics=ics)
 

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