50.19.2 problem 1(b)

Internal problem ID [8110]
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)
Date solved : Monday, January 27, 2025 at 03:43:55 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

Using series method with expansion around

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

✓ Solution by Maple

Time used: 0.016 (sec). Leaf size: 374

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);
 

\[ y = c_{1} x^{-\sqrt {2}} \left (1+\frac {\sqrt {2}}{-1+2 \sqrt {2}} x +\frac {\sqrt {2}}{\left (1-2 \sqrt {2}\right ) \left (\sqrt {2}-1\right )} x^{2}+\frac {6 \sqrt {2}-8}{57 \sqrt {2}-81} x^{3}+\frac {-49 \sqrt {2}+69}{1104-780 \sqrt {2}} x^{4}+\frac {293 \sqrt {2}-414}{6108 \sqrt {2}-8640} x^{5}+\frac {-2757 \sqrt {2}+3898}{114408-80892 \sqrt {2}} x^{6}+\frac {1}{126} \frac {77567 \sqrt {2}-109686}{\left (-1+2 \sqrt {2}\right ) \left (\sqrt {2}-1\right ) \left (-3+2 \sqrt {2}\right ) \left (\sqrt {2}-2\right ) \left (-5+2 \sqrt {2}\right ) \left (-3+\sqrt {2}\right ) \left (-7+2 \sqrt {2}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right )+c_{2} x^{\sqrt {2}} \left (1+\frac {\sqrt {2}}{1+2 \sqrt {2}} x +\frac {\sqrt {2}}{5+3 \sqrt {2}} x^{2}+\frac {6 \sqrt {2}+8}{57 \sqrt {2}+81} x^{3}+\frac {49 \sqrt {2}+69}{1104+780 \sqrt {2}} x^{4}+\frac {293 \sqrt {2}+414}{6108 \sqrt {2}+8640} x^{5}+\frac {2757 \sqrt {2}+3898}{114408+80892 \sqrt {2}} x^{6}+\frac {1}{126} \frac {77567 \sqrt {2}+109686}{\left (1+2 \sqrt {2}\right ) \left (1+\sqrt {2}\right ) \left (3+2 \sqrt {2}\right ) \left (2+\sqrt {2}\right ) \left (5+2 \sqrt {2}\right ) \left (3+\sqrt {2}\right ) \left (7+2 \sqrt {2}\right )} x^{7}+\operatorname {O}\left (x^{8}\right )\right ) \]

✓ Solution by Mathematica

Time used: 0.015 (sec). Leaf size: 10352

AsymptoticDSolveValue[x^2*(x^2-1)*D[y[x],{x,2}]-x*(1-x)*D[y[x],x]+2*y[x]==0,y[x],{x,0,"8"-1}]
 

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