Internal problem ID [5717]
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 4.6. Gauss’s Hypergeometric
Equation. Page 187
Problem number: 2(a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
Solve \begin {gather*} \boxed {x \left (1-x \right ) y^{\prime \prime }+\left (\frac {3}{2}-2 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: 0.031 (sec). Leaf size: 40
Order:=8; dsolve(x*(1-x)*diff(y(x),x$2)+(3/2-2*x)*diff(y(x),x)+2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} \left (1-\frac {4}{3} x +\mathrm {O}\left (x^{8}\right )\right ) \sqrt {x}+c_{1} \left (1-\frac {9}{2} x +\frac {15}{8} x^{2}+\frac {7}{16} x^{3}+\frac {27}{128} x^{4}+\frac {33}{256} x^{5}+\frac {91}{1024} x^{6}+\frac {135}{2048} x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 71
AsymptoticDSolveValue[x*(1-x)*y''[x]+(3/2-2*x)*y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_2 \left (\frac {135 x^7}{2048}+\frac {91 x^6}{1024}+\frac {33 x^5}{256}+\frac {27 x^4}{128}+\frac {7 x^3}{16}+\frac {15 x^2}{8}-\frac {9 x}{2}+1\right )}{\sqrt {x}}+c_1 \left (1-\frac {4 x}{3}\right ) \]