Internal problem ID [1401]
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: 49.
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 }-x \left (9 x^{2}+1\right ) y^{\prime }+\left (25 x^{2}+1\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: 45
Order:=6; dsolve(x^2*(1+x^2)*diff(y(x),x$2)-x*(1+9*x^2)*diff(y(x),x)+(1+25*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-4 x^{2}+x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (6 x^{2}-3 x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 49
AsymptoticDSolveValue[x^2*(1+x^2)*y''[x]-x*(1+9*x^2)*y'[x]+(1+25*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (x^4-4 x^2+1\right )+c_2 \left (x \left (6 x^2-3 x^4\right )+x \left (x^4-4 x^2+1\right ) \log (x)\right ) \]