76.14.2 problem 30

Internal problem ID [17569]
Book : Differential equations. An introduction to modern methods and applications. James Brannan, William E. Boyce. Third edition. Wiley 2015
Section : Chapter 4. Second order linear equations. Section 4.4 (Mechanical and electrical vibration). Problems at page 250
Problem number : 30
Date solved : Monday, March 31, 2025 at 04:17:50 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }+\frac {y^{\prime }}{4}+2 y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.096 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x)+1/4*diff(y(x),x)+2*y(x) = 0; 
ic:=y(0) = 0, D(y)(0) = 2; 
dsolve([ode,ic],y(x), singsol=all);
 

\[ y = \frac {16 \sqrt {127}\, {\mathrm e}^{-\frac {x}{8}} \sin \left (\frac {\sqrt {127}\, x}{8}\right )}{127} \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 30

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

\[ y(x)\to \frac {16 e^{-x/8} \sin \left (\frac {\sqrt {127} x}{8}\right )}{\sqrt {127}} \]

✓ Sympy. Time used: 0.194 (sec). Leaf size: 26

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(2*y(x) + Derivative(y(x), x)/4 + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 0, Subs(Derivative(y(x), x), x, 0): 2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {16 \sqrt {127} e^{- \frac {x}{8}} \sin {\left (\frac {\sqrt {127} x}{8} \right )}}{127} \]