Internal problem ID [1381]
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: 29.
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 }-3 x \left (-x^{2}+1\right ) y^{\prime }+4 y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.009 (sec). Leaf size: 51
Order:=6; dsolve(x^2*(1+x^2)*diff(y(x),x$2)-3*x*(1-x^2)*diff(y(x),x)+4*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-2 x^{2}+3 x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (\frac {1}{2} x^{2}-x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) x^{2} \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 61
AsymptoticDSolveValue[x^2*(1+x^2)*y''[x]-3*x*(1-x^2)*y'[x]+4*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (3 x^4-2 x^2+1\right ) x^2+c_2 \left (\left (\frac {x^2}{2}-x^4\right ) x^2+\left (3 x^4-2 x^2+1\right ) x^2 \log (x)\right ) \]