36.6.6 problem 6

Internal problem ID [6367]
Book : Fundamentals of Differential Equations. By Nagle, Saff and Snider. 9th edition. Boston. Pearson 2018.
Section : Chapter 8, Series solutions of differential equations. Section 8.4. page 449
Problem number : 6
Date solved : Monday, January 27, 2025 at 01:58:17 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x^{3}+1\right ) y^{\prime \prime }-x y^{\prime }+2 x^{2} y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 76

Order:=6; 
dsolve((1+x^3)*diff(y(x),x$2)-x*diff(y(x),x)+2*x^2*y(x)=0,y(x),type='series',x=1);
 

\[ y = \left (1-\frac {\left (x -1\right )^{2}}{2}-\frac {\left (x -1\right )^{3}}{6}+\frac {7 \left (x -1\right )^{4}}{48}+\frac {7 \left (x -1\right )^{5}}{240}\right ) y \left (1\right )+\left (x -1+\frac {\left (x -1\right )^{2}}{4}-\frac {\left (x -1\right )^{3}}{6}-\frac {\left (x -1\right )^{4}}{8}+\frac {\left (x -1\right )^{5}}{12}\right ) y^{\prime }\left (1\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 78

AsymptoticDSolveValue[(1+x^3)*D[y[x],{x,2}]-x*D[y[x],x]+2*x*y[x]==0,y[x],{x,1,"6"-1}]
 

\[ y(x)\to c_1 \left (-\frac {1}{20} (x-1)^5+\frac {1}{8} (x-1)^4-\frac {1}{2} (x-1)^2+1\right )+c_2 \left (\frac {19}{240} (x-1)^5-\frac {1}{24} (x-1)^4-\frac {1}{6} (x-1)^3+\frac {1}{4} (x-1)^2+x-1\right ) \]