73.11.38 problem 17.7 (b)

Internal problem ID [15352]
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 : Tuesday, January 28, 2025 at 07:53:40 AM
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*}

✓ Solution by Maple

Time used: 0.023 (sec). Leaf size: 29

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

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

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 35

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

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