Internal problem ID [6270]
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: 22.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _exact, _linear, _homogeneous]]
Solve \begin {gather*} \boxed {x \left (-x^{2}+1\right ) y^{\prime \prime }+5 \left (-x^{2}+1\right ) y^{\prime }-4 x y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.039 (sec). Leaf size: 32
Order:=8; dsolve(x*(1-x^2)*diff(y(x),x$2)+5*(1-x^2)*diff(y(x),x)-4*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \left (1+\frac {1}{3} x^{2}+\frac {1}{6} x^{4}+\frac {1}{10} x^{6}+\mathrm {O}\left (x^{8}\right )\right )+\frac {c_{2} \left (-144+144 x^{2}+\mathrm {O}\left (x^{8}\right )\right )}{x^{4}} \]
✓ Solution by Mathematica
Time used: 0.02 (sec). Leaf size: 42
AsymptoticDSolveValue[x*(1-x^2)*y''[x]+5*(1-x^2)*y'[x]-4*x*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {1}{x^4}-\frac {1}{x^2}\right )+c_2 \left (\frac {x^6}{10}+\frac {x^4}{6}+\frac {x^2}{3}+1\right ) \]