Internal problem ID [6222]
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: 11.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x y^{\prime \prime }-2 \left (2+x \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: 0.041 (sec). Leaf size: 48
Order:=8; dsolve(x*diff(y(x),x$2)-2*(x+2)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{5} \left (1+x +\frac {4}{7} x^{2}+\frac {5}{21} x^{3}+\frac {5}{63} x^{4}+\frac {1}{45} x^{5}+\frac {8}{1485} x^{6}+\frac {4}{3465} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (2880+2880 x +960 x^{2}+128 x^{5}+128 x^{6}+\frac {512}{7} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.09 (sec). Leaf size: 76
AsymptoticDSolveValue[x*y''[x]-2*(x+2)*y'[x]+4*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {2 x^6}{45}+\frac {2 x^5}{45}+\frac {x^2}{3}+x+1\right )+c_2 \left (\frac {8 x^{11}}{1485}+\frac {x^{10}}{45}+\frac {5 x^9}{63}+\frac {5 x^8}{21}+\frac {4 x^7}{7}+x^6+x^5\right ) \]