Internal problem ID [6213]
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: 2.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (1+2 x \right ) y^{\prime \prime }+2 x \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: 0.033 (sec). Leaf size: 53
Order:=8; dsolve(x^2*(1+2*x)*diff(y(x),x$2)+2*x*(1+6*x)*diff(y(x),x)-2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x \left (1-3 x +\frac {42}{5} x^{2}-\frac {112}{5} x^{3}+\frac {288}{5} x^{4}-144 x^{5}+352 x^{6}-\frac {4224}{5} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (12-72 x +288 x^{2}-960 x^{3}+2880 x^{4}-8064 x^{5}+21504 x^{6}-55296 x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{x^{2}} \]
✓ Solution by Mathematica
Time used: 0.103 (sec). Leaf size: 76
AsymptoticDSolveValue[x^2*(1+2*x)*y''[x]+2*x*(1+6*x)*y'[x]-2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (1792 x^4-672 x^3+240 x^2+\frac {1}{x^2}-80 x-\frac {6}{x}+24\right )+c_2 \left (352 x^7-144 x^6+\frac {288 x^5}{5}-\frac {112 x^4}{5}+\frac {42 x^3}{5}-3 x^2+x\right ) \]