Internal problem ID [6227]
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: 16.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (4 x -1\right ) y^{\prime \prime }+x \left (5 x +1\right ) y^{\prime }+3 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 51
Order:=8; dsolve(x^2*(4*x-1)*diff(y(x),x$2)+x*(5*x+1)*diff(y(x),x)+3*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{3} \left (1+\frac {39}{5} x +\frac {221}{5} x^{2}+221 x^{3}+\frac {16575}{16} x^{4}+\frac {224315}{48} x^{5}+\frac {493493}{24} x^{6}+\frac {711399}{8} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-144+144 x +270 x^{4}+2106 x^{5}+11934 x^{6}+59670 x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.138 (sec). Leaf size: 80
AsymptoticDSolveValue[x^2*(4*x-1)*y''[x]+x*(5*x+1)*y'[x]+3*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (-\frac {663 x^5}{8}-\frac {117 x^4}{8}-\frac {15 x^3}{8}+\frac {1}{x}-1\right )+c_2 \left (\frac {493493 x^9}{24}+\frac {224315 x^8}{48}+\frac {16575 x^7}{16}+221 x^6+\frac {221 x^5}{5}+\frac {39 x^4}{5}+x^3\right ) \]