73.11.38 problem 17.7 (b)

Internal problem ID [15281]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 17. Second order Homogeneous equations with constant coefficients. Additional exercises page 334
Problem number : 17.7 (b)
Date solved : Monday, March 31, 2025 at 01:33:21 PM
CAS classification : [[_2nd_order, _missing_x]]

\begin{align*} y^{\prime \prime }-y^{\prime }+\left (\frac {1}{4}+4 \pi ^{2}\right ) y&=0 \end{align*}

With initial conditions

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

✓ Maple. Time used: 0.072 (sec). Leaf size: 25

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

\[ y = {\mathrm e}^{\frac {x}{2}} \left (-\frac {\sin \left (2 \pi x \right )}{2 \pi }+\cos \left (2 \pi x \right )\right ) \]

✓ Mathematica. Time used: 0.02 (sec). Leaf size: 35

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

\[ y(x)\to \frac {e^{x/2} (2 \pi \cos (2 \pi x)-\sin (2 \pi x))}{2 \pi } \]

✓ Sympy. Time used: 0.212 (sec). Leaf size: 24

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((1/4 + 4*pi**2)*y(x) - Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): -1/2} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \left (- \frac {\sin {\left (2 \pi x \right )}}{2 \pi } + \cos {\left (2 \pi x \right )}\right ) e^{\frac {x}{2}} \]