Internal problem ID [1407]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.6 THE METHOD OF FROBENIUS
II. Exercises 7.6. Page 374
Problem number: 60.
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}+2\right ) y^{\prime \prime }-x \left (3 x^{2}+2\right ) y^{\prime }+\left (-x^{2}+2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.01 (sec). Leaf size: 33
Order:=6; dsolve(x^2*(2-x^2)*diff(y(x),x$2)-x*(2+3*x^2)*diff(y(x),x)+(2-x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (1+\frac {1}{2} x^{2}+\frac {1}{4} x^{4}\right ) x \left (\ln \relax (x ) c_{2}+c_{1}\right )+O\left (x^{6}\right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 46
AsymptoticDSolveValue[x^2*(2-x^2)*y''[x]-x*(2+3*x^2)*y'[x]+(2-x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (\frac {x^4}{4}+\frac {x^2}{2}+1\right )+c_2 x \left (\frac {x^4}{4}+\frac {x^2}{2}+1\right ) \log (x) \]