Internal problem ID [1386]
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: 34.
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 (x^{2}+4\right ) y^{\prime \prime }+3 x \left (3 x^{2}+8\right ) y^{\prime }+\left (-9 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: 0.011 (sec). Leaf size: 51
Order:=6; dsolve(4*x^2*(4+x^2)*diff(y(x),x$2)+3*x*(8+3*x^2)*diff(y(x),x)+(1-9*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1+\frac {5}{32} x^{2}-\frac {15}{2048} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (-\frac {13}{64} x^{2}+\frac {13}{8192} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}}{x^{\frac {1}{4}}} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 77
AsymptoticDSolveValue[4*x^2*(4+x^2)*y''[x]+3*x*(8+3*x^2)*y'[x]+(1-9*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to \frac {c_1 \left (-\frac {15 x^4}{2048}+\frac {5 x^2}{32}+1\right )}{\sqrt [4]{x}}+c_2 \left (\frac {\frac {13 x^4}{8192}-\frac {13 x^2}{64}}{\sqrt [4]{x}}+\frac {\left (-\frac {15 x^4}{2048}+\frac {5 x^2}{32}+1\right ) \log (x)}{\sqrt [4]{x}}\right ) \]