89.27.1 problem 1

Internal problem ID [24883]
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 : 1
Date solved : Thursday, October 02, 2025 at 10:49:03 PM
CAS classification : system_of_ODEs

\begin{align*} v^{\prime }\left (x \right )-2 v \left (x \right )+2 w^{\prime }\left (x \right )&=2-4 \,{\mathrm e}^{2 x}\\ 2 v^{\prime }\left (x \right )-3 v \left (x \right )+3 w^{\prime }\left (x \right )-w \left (x \right )&=0 \end{align*}

✓ Maple. Time used: 0.076 (sec). Leaf size: 45

ode:=[diff(v(x),x)-2*v(x)+2*diff(w(x),x) = 2-4*exp(2*x), 2*diff(v(x),x)-3*v(x)+3*diff(w(x),x)-w(x) = 0]; 
dsolve(ode);
 

\begin{align*} v \left (x \right ) &= {\mathrm e}^{x} c_2 +{\mathrm e}^{-2 x} c_1 +5 \,{\mathrm e}^{2 x}-1 \\ w \left (x \right ) &= \frac {{\mathrm e}^{x} c_2}{2}-{\mathrm e}^{-2 x} c_1 -{\mathrm e}^{2 x}+3 \\ \end{align*}

✓ Mathematica. Time used: 0.051 (sec). Leaf size: 93

ode={D[v[x],x]-2*v[x]+2*D[w[x],x]==2-4*Exp[2*x],2*D[v[x],x]-3*v[x]+3*D[w[x],x]-w[x]==0}; 
ic={}; 
DSolve[{ode,ic},{v[x],w[x]},x,IncludeSingularSolutions->True]
 

\begin{align*} v(x)&\to \frac {1}{3} e^{-2 x} \left (-3 e^{2 x}+15 e^{4 x}+2 (c_1+c_2) e^{3 x}+c_1-2 c_2\right )\\ w(x)&\to \frac {1}{3} e^{-2 x} \left (9 e^{2 x}-3 e^{4 x}+(c_1+c_2) e^{3 x}-c_1+2 c_2\right ) \end{align*}

✓ Sympy. Time used: 0.139 (sec). Leaf size: 44

from sympy import * 
x = symbols("x") 
v = Function("v") 
w = Function("w") 
ode=[Eq(-2*v(x) + 4*exp(2*x) + Derivative(v(x), x) + 2*Derivative(w(x), x) - 2,0),Eq(-3*v(x) - w(x) + 2*Derivative(v(x), x) + 3*Derivative(w(x), x),0)] 
ics = {} 
dsolve(ode,func=[v(x),w(x)],ics=ics)
 

\[ \left [ v{\left (x \right )} = - C_{1} e^{- 2 x} + 2 C_{2} e^{x} + 5 e^{2 x} - 1, \ w{\left (x \right )} = C_{1} e^{- 2 x} + C_{2} e^{x} - e^{2 x} + 3\right ] \]