74.15.59 problem 59

Internal problem ID [16498]
Book : INTRODUCTORY DIFFERENTIAL EQUATIONS. Martha L. Abell, James P. Braselton. Fourth edition 2014. ElScAe. 2014
Section : Chapter 4. Higher Order Equations. Exercises 4.7, page 195
Problem number : 59
Date solved : Tuesday, January 28, 2025 at 09:09:47 AM
CAS classification : [[_Emden, _Fowler], [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

With initial conditions

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

✓ Solution by Maple

Time used: 0.089 (sec). Leaf size: 27

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

\[ y = -\cosh \left (2 \pi \right ) \sin \left (2 \ln \left (x \right )\right )+i \sinh \left (2 \pi \right ) \cos \left (2 \ln \left (x \right )\right ) \]

✓ Solution by Mathematica

Time used: 0.020 (sec). Leaf size: 20

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

\[ y(x)\to i \sinh (2 (\pi +i \log (x))) \]