73.26.12 problem 36.2 (L)

Internal problem ID [15752]
Book : Ordinary Differential Equations. An introduction to the fundamentals. Kenneth B. Howell. second edition. CRC Press. FL, USA. 2020
Section : Chapter 36. The big theorem on the the Frobenius method. Additional Exercises. page 739
Problem number : 36.2 (L)
Date solved : Tuesday, January 28, 2025 at 08:06:37 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 }+3 y&=0 \end{align*}

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.052 (sec). Leaf size: 46

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

\[ y = c_{1} \sqrt {x -1}\, \left (1+\frac {11}{12} \left (x -1\right )+\frac {11}{160} \left (x -1\right )^{2}-\frac {143}{13440} \left (x -1\right )^{3}+\frac {5291}{1935360} \left (x -1\right )^{4}-\frac {11063}{12902400} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right )+c_{2} \left (1+3 \left (x -1\right )+\left (x -1\right )^{2}-\frac {1}{15} \left (x -1\right )^{3}+\frac {1}{70} \left (x -1\right )^{4}-\frac {13}{3150} \left (x -1\right )^{5}+\operatorname {O}\left (\left (x -1\right )^{6}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.005 (sec). Leaf size: 101

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

\[ y(x)\to c_1 \left (-\frac {11063 (x-1)^5}{12902400}+\frac {5291 (x-1)^4}{1935360}-\frac {143 (x-1)^3}{13440}+\frac {11}{160} (x-1)^2+\frac {11 (x-1)}{12}+1\right ) \sqrt {x-1}+c_2 \left (-\frac {13 (x-1)^5}{3150}+\frac {1}{70} (x-1)^4-\frac {1}{15} (x-1)^3+(x-1)^2+3 (x-1)+1\right ) \]