36.5.6 problem 6

Internal problem ID [6350]
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 : 6
Date solved : Monday, January 27, 2025 at 01:57:52 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.004 (sec). Leaf size: 59

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

\[ y = \left (1+\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{12} x^{4}-\frac {1}{15} x^{5}\right ) y \left (0\right )+\left (x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}-\frac {1}{12} x^{4}+\frac {1}{60} x^{5}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.002 (sec). Leaf size: 70

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

\[ y(x)\to c_1 \left (-\frac {x^5}{15}+\frac {x^4}{12}-\frac {x^3}{6}+\frac {x^2}{2}+1\right )+c_2 \left (\frac {x^5}{60}-\frac {x^4}{12}+\frac {x^3}{6}+\frac {x^2}{2}+x\right ) \]