Internal problem ID [4896]
Book: ADVANCED ENGINEERING MATHEMATICS. ERWIN KREYSZIG, HERBERT KREYSZIG,
EDWARD J. NORMINTON. 10th edition. John Wiley USA. 2011
Section: Chapter 5. Series Solutions of ODEs. Special Functions. Problem set 5.3. Extended Power
Series Method: Frobenius Method page 186
Problem number: 16.
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 {1}{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: 0.047 (sec). Leaf size: 36
Order:=6; dsolve(x*(1-x)*diff(y(x),x$2)+(1/2+2*x)*diff(y(x),x)-2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{2} x -\frac {1}{40} x^{2}-\frac {1}{560} x^{3}-\frac {1}{2688} x^{4}-\frac {1}{8448} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (1+4 x +\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 55
AsymptoticDSolveValue[x*(1-x)*y''[x]+(1/2+2*x)*y'[x]-2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (-\frac {x^5}{8448}-\frac {x^4}{2688}-\frac {x^3}{560}-\frac {x^2}{40}+\frac {x}{2}+1\right )+c_2 (4 x+1) \]