10.14.13 problem 10

Internal problem ID [1395]
Book : Elementary differential equations and boundary value problems, 10th ed., Boyce and DiPrima
Section : Chapter 5.3, Series Solutions Near an Ordinary Point, Part II. page 269
Problem number : 10
Date solved : Monday, January 27, 2025 at 04:56:41 AM
CAS classification : [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.003 (sec). Leaf size: 65

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

\[ y = \left (1-\frac {\alpha ^{2} x^{2}}{2}+\frac {\alpha ^{2} \left (\alpha ^{2}-4\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (\alpha ^{2}-1\right ) x^{3}}{6}+\frac {\left (\alpha ^{4}-10 \alpha ^{2}+9\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.001 (sec). Leaf size: 88

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

\[ y(x)\to c_2 \left (\frac {\alpha ^4 x^5}{120}-\frac {\alpha ^2 x^5}{12}+\frac {3 x^5}{40}-\frac {\alpha ^2 x^3}{6}+\frac {x^3}{6}+x\right )+c_1 \left (\frac {\alpha ^4 x^4}{24}-\frac {\alpha ^2 x^4}{6}-\frac {\alpha ^2 x^2}{2}+1\right ) \]