Internal problem ID [14016]
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).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Gegenbauer, [_2nd_order, _linear, `_with_symmetry_[0,F(x)]`]]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +3 y=0} \] With the expansion point for the power series method at \(x = 1\).
✓ Solution by Maple
Time used: 0.031 (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);
\[ y \left (x \right ) = c_{1} \sqrt {-1+x}\, \left (1+\frac {11}{12} \left (-1+x \right )+\frac {11}{160} \left (-1+x \right )^{2}-\frac {143}{13440} \left (-1+x \right )^{3}+\frac {5291}{1935360} \left (-1+x \right )^{4}-\frac {11063}{12902400} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right )\right )+c_{2} \left (1+3 \left (-1+x \right )+\left (-1+x \right )^{2}-\frac {1}{15} \left (-1+x \right )^{3}+\frac {1}{70} \left (-1+x \right )^{4}-\frac {13}{3150} \left (-1+x \right )^{5}+\operatorname {O}\left (\left (-1+x \right )^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 101
AsymptoticDSolveValue[(1-x^2)*y''[x]-x*y'[x]+3*y[x]==0,y[x],{x,1,5}]
\[ y(x)\to c_1 \left (-\frac {11063 (x-1)^5}{12902400}+\frac {5291 (x-1)^4}{1935360}-\frac {143 (x-1)^3}{13440}+\frac {11}{160} (x-1)^2+\frac {11 (x-1)}{12}+1\right ) \sqrt {x-1}+c_2 \left (-\frac {13 (x-1)^5}{3150}+\frac {1}{70} (x-1)^4-\frac {1}{15} (x-1)^3+(x-1)^2+3 (x-1)+1\right ) \]