Internal problem ID [6254]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. Miscellaneous Exercises. page 394
Problem number: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (x^{2}+1\right ) y^{\prime \prime }+2 x \left (x^{2}+3\right ) y^{\prime }+6 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: 31
Order:=8; dsolve(x^2*(1+x^2)*diff(y(x),x$2)+2*x*(3+x^2)*diff(y(x),x)+6*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} \left (1-\frac {1}{3} x^{2}+\mathrm {O}\left (x^{8}\right )\right ) x +c_{2} \left (1-3 x^{2}+\mathrm {O}\left (x^{8}\right )\right )}{x^{3}} \]
✓ Solution by Mathematica
Time used: 0.036 (sec). Leaf size: 26
AsymptoticDSolveValue[x^2*(1+x^2)*y''[x]+2*x*(3+x^2)*y'[x]+6*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {1}{x^3}-\frac {3}{x}\right )+c_2 \left (\frac {1}{x^2}-\frac {1}{3}\right ) \]