Internal problem ID [2308]
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.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+4 y^{\prime } x +\left (4 x^{2}-1\right ) y=0} \end {gather*} Given that one solution of the ode is \begin {align*} y_1 &= \frac {\sin \relax (x )}{\sqrt {x}} \end {align*}
✓ Solution by Maple
Time used: 0.086 (sec). Leaf size: 19
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)],y(x), singsol=all)
\[ y \relax (x ) = \frac {c_{1} \sin \relax (x )}{\sqrt {x}}+\frac {c_{2} \cos \relax (x )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 39
DSolve[4*x^2*y''[x]+4*x*y'[x]+(4*x^2-1)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to \frac {e^{-i x} \left (2 c_1-i c_2 e^{2 i x}\right )}{2 \sqrt {x}} \\ \end{align*}