Internal problem ID [6225]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.8 Indicial Equation with Difference of Roots a
Positive Integer: Nonlogarithmic Case. Exercises page 380
Problem number: 14.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x \left (-2 x +1\right ) y^{\prime \prime }-2 \left (x +2\right ) y^{\prime }+8 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.079 (sec). Leaf size: 48
Order:=8; dsolve(x*(1-2*x)*diff(y(x),x$2)-2*(2+x)*diff(y(x),x)+8*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{5} \left (1+7 x +32 x^{2}+120 x^{3}+400 x^{4}+1232 x^{5}+3584 x^{6}+9984 x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (2880+5760 x +5760 x^{2}+92160 x^{5}+645120 x^{6}+2949120 x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.18 (sec). Leaf size: 64
AsymptoticDSolveValue[x*(1-2*x)*y''[x]-2*(2+x)*y'[x]+8*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (224 x^6+32 x^5+2 x^2+2 x+1\right )+c_2 \left (3584 x^{11}+1232 x^{10}+400 x^9+120 x^8+32 x^7+7 x^6+x^5\right ) \]