Internal problem ID [1403]
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: 51.
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 }+\left (-x^{2}+1\right ) y^{\prime }-8 y x=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: 41
Order:=6; dsolve(x*(1+x^2)*diff(y(x),x$2)+(1-x^2)*diff(y(x),x)-8*x*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (-\frac {3}{2} x^{2}-\frac {3}{2} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}+\left (1+2 x^{2}+x^{4}+\mathrm {O}\left (x^{6}\right )\right ) \left (\ln \relax (x ) c_{2}+c_{1}\right ) \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 48
AsymptoticDSolveValue[x*(1+x^2)*y''[x]+(1-x^2)*y'[x]-8*x*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (x^4+2 x^2+1\right )+c_2 \left (-\frac {3 x^4}{2}-\frac {3 x^2}{2}+\left (x^4+2 x^2+1\right ) \log (x)\right ) \]