Internal problem ID [1388]
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: 36.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} \left (4 x^{2}+1\right ) y^{\prime \prime }+32 y^{\prime } x^{3}+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 51
Order:=6; dsolve(4*x^2*(1+4*x^2)*diff(y(x),x$2)+32*x^3*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \sqrt {x}\, \left (\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1-\frac {3}{4} x^{2}+\frac {105}{64} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (-\frac {5}{4} x^{2}+\frac {389}{128} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) \]
✓ Solution by Mathematica
Time used: 0.005 (sec). Leaf size: 77
AsymptoticDSolveValue[4*x^2*(1+4*x^2)*y''[x]+32*x^3*y'[x]+y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {105 x^4}{64}-\frac {3 x^2}{4}+1\right )+c_2 \left (\sqrt {x} \left (\frac {389 x^4}{128}-\frac {5 x^2}{4}\right )+\sqrt {x} \left (\frac {105 x^4}{64}-\frac {3 x^2}{4}+1\right ) \log (x)\right ) \]