Internal problem ID [4895]
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: 15.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {2 x \left (1-x \right ) y^{\prime \prime }-\left (1+6 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: 0.025 (sec). Leaf size: 44
Order:=6; dsolve(2*x*(1-x)*diff(y(x),x$2)-(1+6*x)*diff(y(x),x)-2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {3}{2}} \left (1+\frac {5}{2} x +\frac {35}{8} x^{2}+\frac {105}{16} x^{3}+\frac {1155}{128} x^{4}+\frac {3003}{256} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (1-2 x -8 x^{2}-16 x^{3}-\frac {128}{5} x^{4}-\frac {256}{7} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 79
AsymptoticDSolveValue[2*x*(1-x)*y''[x]-(1+6*x)*y'[x]-2*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (-\frac {256 x^5}{7}-\frac {128 x^4}{5}-16 x^3-8 x^2-2 x+1\right )+c_1 \left (\frac {3003 x^5}{256}+\frac {1155 x^4}{128}+\frac {105 x^3}{16}+\frac {35 x^2}{8}+\frac {5 x}{2}+1\right ) x^{3/2} \]