4.8.18 problem 24

Internal problem ID [1290]
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 : 24
Date solved : Tuesday, September 30, 2025 at 04:32:13 AM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} 5 u^{\prime \prime }+2 u^{\prime }+7 u&=0 \end{align*}

With initial conditions

\begin{align*} u \left (0\right )&=2 \\ u^{\prime }\left (0\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.104 (sec). Leaf size: 32

ode:=5*diff(diff(u(x),x),x)+2*diff(u(x),x)+7*u(x) = 0; 
ic:=[u(0) = 2, D(u)(0) = 1]; 
dsolve([ode,op(ic)],u(x), singsol=all);
 

\[ u = \frac {{\mathrm e}^{-\frac {x}{5}} \left (7 \sqrt {34}\, \sin \left (\frac {\sqrt {34}\, x}{5}\right )+68 \cos \left (\frac {\sqrt {34}\, x}{5}\right )\right )}{34} \]

✓ Mathematica. Time used: 0.015 (sec). Leaf size: 48

ode=5*D[u[x],{x,2}]+2*D[u[x],x]+7*u[x]==0; 
ic={u[0]==2,Derivative[1][u][0]==1}; 
DSolve[{ode,ic},u[x],x,IncludeSingularSolutions->True]
 

\begin{align*} u(x)&\to \frac {1}{34} e^{-x/5} \left (7 \sqrt {34} \sin \left (\frac {\sqrt {34} x}{5}\right )+68 \cos \left (\frac {\sqrt {34} x}{5}\right )\right ) \end{align*}

✓ Sympy. Time used: 0.129 (sec). Leaf size: 37

from sympy import * 
x = symbols("x") 
u = Function("u") 
ode = Eq(7*u(x) + 2*Derivative(u(x), x) + 5*Derivative(u(x), (x, 2)),0) 
ics = {u(0): 2, Subs(Derivative(u(x), x), x, 0): 1} 
dsolve(ode,func=u(x),ics=ics)
 

\[ u{\left (x \right )} = \left (\frac {7 \sqrt {34} \sin {\left (\frac {\sqrt {34} x}{5} \right )}}{34} + 2 \cos {\left (\frac {\sqrt {34} x}{5} \right )}\right ) e^{- \frac {x}{5}} \]