Internal problem ID [5689]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR
POINTS. Page 175
Problem number: 1(b).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (x^{2}-1\right ) y^{\prime \prime }-x \left (1-x \right ) y^{\prime }+2 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 654
Order:=8; dsolve(x^2*(x^2-1)*diff(y(x),x$2)-x*(1-x)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{-\sqrt {2}} \left (1+\frac {\sqrt {2}}{-1+2 \sqrt {2}} x +\frac {\sqrt {2}}{-5+3 \sqrt {2}} x^{2}+\frac {6 \sqrt {2}-8}{57 \sqrt {2}-81} x^{3}+\frac {-49 \sqrt {2}+69}{1104-780 \sqrt {2}} x^{4}+\frac {293 \sqrt {2}-414}{6108 \sqrt {2}-8640} x^{5}+\frac {-2757 \sqrt {2}+3898}{114408-80892 \sqrt {2}} x^{6}+\frac {77567 \sqrt {2}-109686}{2782710 \sqrt {2}-3935484} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x^{\sqrt {2}} \left (1+\frac {\sqrt {2}}{1+2 \sqrt {2}} x +\frac {\sqrt {2}}{5+3 \sqrt {2}} x^{2}+\frac {6 \sqrt {2}+8}{57 \sqrt {2}+81} x^{3}+\frac {49 \sqrt {2}+69}{1104+780 \sqrt {2}} x^{4}+\frac {293 \sqrt {2}+414}{6108 \sqrt {2}+8640} x^{5}+\frac {2757 \sqrt {2}+3898}{114408+80892 \sqrt {2}} x^{6}+\frac {77567 \sqrt {2}+109686}{2782710 \sqrt {2}+3935484} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 10352
AsymptoticDSolveValue[x^2*(x^2-1)*y''[x]-x*(1-x)*y'[x]+2*y[x]==0,y[x],{x,0,7}]
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