Internal problem ID [6174]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.4 Indicial Equation with Difference of Roots
Nonintegral. Exercises page 365
Problem number: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries], [_2nd_order, _linear, _with_symmetry_[0,F(x)]]]
Solve \begin {gather*} \boxed {x \left (4-x \right ) y^{\prime \prime }+\left (-x +2\right ) y^{\prime }+4 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 42
Order:=8; dsolve(x*(4-x)*diff(y(x),x$2)+(2-x)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1-\frac {5}{8} x +\frac {7}{128} x^{2}+\frac {3}{1024} x^{3}+\frac {11}{32768} x^{4}+\frac {13}{262144} x^{5}+\frac {35}{4194304} x^{6}+\frac {51}{33554432} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (1-2 x +\frac {1}{2} x^{2}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.047 (sec). Leaf size: 76
AsymptoticDSolveValue[x*(4-x)*y''[x]+(2-x)*y'[x]+4*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {x^2}{2}-2 x+1\right )+c_1 \sqrt {x} \left (\frac {51 x^7}{33554432}+\frac {35 x^6}{4194304}+\frac {13 x^5}{262144}+\frac {11 x^4}{32768}+\frac {3 x^3}{1024}+\frac {7 x^2}{128}-\frac {5 x}{8}+1\right ) \]