78.11.6 problem 6

Internal problem ID [18277]
Book : DIFFERENTIAL EQUATIONS WITH APPLICATIONS AND HISTORICAL NOTES by George F. Simmons. 3rd edition. 2017. CRC press, Boca Raton FL.
Section : Chapter 3. Second order linear equations. Section 16. The Use of a Known Solution to find Another. Problems at page 121
Problem number : 6
Date solved : Tuesday, January 28, 2025 at 11:45:03 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime }+x y^{\prime }+\left (x^{2}-\frac {1}{4}\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: 0.013 (sec). Leaf size: 17

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

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

✓ Solution by Mathematica

Time used: 0.018 (sec). Leaf size: 33

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

\[ y(x)\to c_1 x-\frac {1}{2} c_2 (x \log (1-x)-x \log (x+1)+2) \]