49.15.10 problem 6

Internal problem ID [7696]
Book : An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY 1961
Section : Chapter 3. Linear equations with variable coefficients. Page 130
Problem number : 6
Date solved : Monday, January 27, 2025 at 03:09:58 PM
CAS classification : [_Gegenbauer]

\begin{align*} \left (-x^{2}+1\right ) y^{\prime \prime }-2 x y^{\prime }+\alpha \left (\alpha +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: 81

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

\[ y = \left (1-\frac {\alpha \left (\alpha +1\right ) x^{2}}{2}+\frac {\alpha \left (\alpha ^{3}+2 \alpha ^{2}-5 \alpha -6\right ) x^{4}}{24}\right ) y \left (0\right )+\left (x -\frac {\left (\alpha ^{2}+\alpha -2\right ) x^{3}}{6}+\frac {\left (\alpha ^{4}+2 \alpha ^{3}-13 \alpha ^{2}-14 \alpha +24\right ) x^{5}}{120}\right ) y^{\prime }\left (0\right )+O\left (x^{6}\right ) \]

✓ Solution by Mathematica

Time used: 0.003 (sec). Leaf size: 127

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

\[ y(x)\to c_2 \left (\frac {1}{60} \left (-\alpha ^2-\alpha \right ) x^5-\frac {1}{120} \left (-\alpha ^2-\alpha \right ) \left (\alpha ^2+\alpha \right ) x^5-\frac {1}{10} \left (\alpha ^2+\alpha \right ) x^5+\frac {x^5}{5}-\frac {1}{6} \left (\alpha ^2+\alpha \right ) x^3+\frac {x^3}{3}+x\right )+c_1 \left (\frac {1}{24} \left (\alpha ^2+\alpha \right )^2 x^4-\frac {1}{4} \left (\alpha ^2+\alpha \right ) x^4-\frac {1}{2} \left (\alpha ^2+\alpha \right ) x^2+1\right ) \]