36.5.9 problem 9

Internal problem ID [6353]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.3. page 443
Problem number : 9
Date solved : Monday, January 27, 2025 at 01:57:56 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \sin \left (x \right ) y^{\prime \prime }-y \ln \left (x \right )&=0 \end{align*}

Using series method with expansion around

\begin{align*} 1 \end{align*}

✓ Solution by Maple

Time used: 0.007 (sec). Leaf size: 90

Order:=6; 
dsolve(sin(x)*diff(y(x),x$2)-ln(x)*y(x)=0,y(x),type='series',x=1);
 

\[ y = \left (1+\frac {\csc \left (1\right ) \left (x -1\right )^{3}}{6}-\frac {\csc \left (1\right ) \left (2 \cot \left (1\right )+1\right ) \left (x -1\right )^{4}}{24}+\frac {\left (11+\cos \left (2\right )+3 \sin \left (2\right )\right ) \csc \left (1\right )^{3} \left (x -1\right )^{5}}{240}\right ) y \left (1\right )+\left (x -1+\frac {\csc \left (1\right ) \left (x -1\right )^{4}}{12}-\frac {\left (\sin \left (2\right )+\sin \left (1\right )^{2}\right ) \csc \left (1\right )^{3} \left (x -1\right )^{5}}{40}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 107

AsymptoticDSolveValue[Sin[x]*D[y[x],{x,2}]-Log[x]*y[x]==0,y[x],{x,1,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {1}{6} (x-1)^3 \csc (1)-\frac {1}{120} (x-1)^5 \left (-5 \csc (1)-6 \cot ^2(1) \csc (1)-3 \cot (1) \csc (1)\right )-\frac {1}{12} (x-1)^4 \left (\frac {\csc (1)}{2}+\cot (1) \csc (1)\right )+1\right )+c_2 \left (x+\frac {1}{12} (x-1)^4 \csc (1)-\frac {1}{20} (x-1)^5 \left (\frac {\csc (1)}{2}+\cot (1) \csc (1)\right )-1\right ) \]