85.83.2 problem 1 (b)

Internal problem ID [22999]
Book : Applied Differential Equations. By Murray R. Spiegel. 3rd edition. 1980. Pearson. ISBN 978-0130400970
Section : Chapter 10. Systems of differential equations and their applications. A Exercises at page 444
Problem number : 1 (b)
Date solved : Thursday, October 02, 2025 at 09:17:25 PM
CAS classification : system_of_ODEs

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

✓ Maple. Time used: 0.054 (sec). Leaf size: 36

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

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 74

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

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

✓ Sympy. Time used: 0.099 (sec). Leaf size: 37

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

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