Internal problem ID [6226]
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: 15.
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 }+\left (x^{3}-1\right ) y^{\prime }+x^{2} y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 32
Order:=8; dsolve(x*diff(y(x),x$2)+(x^3-1)*diff(y(x),x)+x^2*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{2} \left (1-\frac {1}{5} x^{3}+\frac {1}{40} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (-2+\frac {2}{3} x^{3}-\frac {1}{9} x^{6}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.016 (sec). Leaf size: 44
AsymptoticDSolveValue[x*y''[x]+(x^3-1)*y'[x]+x^2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {x^6}{18}-\frac {x^3}{3}+1\right )+c_2 \left (\frac {x^8}{40}-\frac {x^5}{5}+x^2\right ) \]