Internal problem ID [5589]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 2. Second-Order Linear Equations. Section 2.4. THE USE OF A KNOWN
SOLUTION TO FIND ANOTHER. Page 74
Problem number: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\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: 0.112 (sec). Leaf size: 19
dsolve([x^2*diff(y(x),x$2)+x*diff(y(x),x)+(x^2-1/4)*y(x)=0,x^(-1/2)*sin(x)],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: 0.014 (sec). Leaf size: 39
DSolve[x^2*y''[x]+x*y'[x]+(x^2-1/4)*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*}