20.11.6 problem Problem 6

Internal problem ID [3788]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.9, Reduction of Order. page 572
Problem number : Problem 6
Date solved : Monday, January 27, 2025 at 08:02:19 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using reduction of order method given that one solution is

\begin{align*} y&=\frac {\sin \left (x \right )}{\sqrt {x}} \end{align*}

✓ Solution by Maple

Time used: 0.050 (sec). Leaf size: 17

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

\[ y \left (x \right ) = \frac {c_{1} \sin \left (x \right )+c_{2} \cos \left (x \right )}{\sqrt {x}} \]

✓ Solution by Mathematica

Time used: 0.032 (sec). Leaf size: 39

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

\[ y(x)\to \frac {e^{-i x} \left (2 c_1-i c_2 e^{2 i x}\right )}{2 \sqrt {x}} \]