36.5.3 problem 3

Internal problem ID [6347]
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 : 3
Date solved : Monday, January 27, 2025 at 01:57:47 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

\begin{align*} 0 \end{align*}

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 54

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

\[ y = \left (1+\frac {1}{12} x^{3}+\frac {1}{48} x^{4}+\frac {1}{80} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{8} x^{4}+\frac {1}{16} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 63

AsymptoticDSolveValue[(x^2-2)*D[y[x],{x,2}]+2*D[y[x],x]+Sin[x]*y[x]==0,y[x],{x,0,"6"-1}]
 

\[ y(x)\to c_1 \left (\frac {x^5}{80}+\frac {x^4}{48}+\frac {x^3}{12}+1\right )+c_2 \left (\frac {x^5}{16}+\frac {x^4}{8}+\frac {x^3}{6}+\frac {x^2}{2}+x\right ) \]