4.8.17 problem 23

Internal problem ID [1289]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 3, Second order linear equations, 3.3 Complex Roots of the Characteristic Equation , page 164
Problem number : 23
Date solved : Tuesday, September 30, 2025 at 04:32:12 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} u^{\prime \prime }-u^{\prime }+2 u&=0 \end{align*}

With initial conditions

\begin{align*} u \left (0\right )&=2 \\ u^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.108 (sec). Leaf size: 31

ode:=diff(diff(u(x),x),x)-diff(u(x),x)+2*u(x) = 0; 
ic:=[u(0) = 2, D(u)(0) = 0]; 
dsolve([ode,op(ic)],u(x), singsol=all);
 

\[ u = -\frac {2 \,{\mathrm e}^{\frac {x}{2}} \left (\sqrt {7}\, \sin \left (\frac {\sqrt {7}\, x}{2}\right )-7 \cos \left (\frac {\sqrt {7}\, x}{2}\right )\right )}{7} \]

✓ Mathematica. Time used: 0.01 (sec). Leaf size: 19

ode=D[u[x],{x,2}]+4*D[u[x],x]+5*u[x]==0; 
ic={u[0]==2,Derivative[1][u][0]==0}; 
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
 

\begin{align*} u(x)&\to 2 e^{-2 x} (2 \sin (x)+\cos (x)) \end{align*}

✓ Sympy. Time used: 0.125 (sec). Leaf size: 37

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

\[ u{\left (x \right )} = \left (- \frac {2 \sqrt {7} \sin {\left (\frac {\sqrt {7} x}{2} \right )}}{7} + 2 \cos {\left (\frac {\sqrt {7} x}{2} \right )}\right ) e^{\frac {x}{2}} \]