Internal problem ID [1444]
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
III. Exercises 7.7. Page 389
Problem number: 28.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {4 x^{2} y^{\prime \prime }+2 x \left (x^{2}+8\right ) y^{\prime }+\left (3 x^{2}+5\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.013 (sec). Leaf size: 51
Order:=6; dsolve(4*x^2*diff(y(x),x$2)+2*x*(8+x^2)*diff(y(x),x)+(5+3*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} \left (1-\frac {1}{16} x^{2}+\frac {1}{256} x^{4}+\mathrm {O}\left (x^{6}\right )\right ) x^{2}+c_{2} \left (\ln \relax (x ) \left (-\frac {1}{2} x^{2}+\frac {1}{32} x^{4}+\mathrm {O}\left (x^{6}\right )\right )+\left (-2+\frac {1}{2} x^{2}-\frac {3}{128} x^{4}+\mathrm {O}\left (x^{6}\right )\right )\right )}{x^{\frac {5}{2}}} \]
✓ Solution by Mathematica
Time used: 0.012 (sec). Leaf size: 72
AsymptoticDSolveValue[4*x^2*y''[x]+2*x*(8+x^2)*y'[x]+(5+3*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^{7/2}}{256}-\frac {x^{3/2}}{16}+\frac {1}{\sqrt {x}}\right )+c_1 \left (\frac {5 x^4-96 x^2+256}{256 x^{5/2}}-\frac {\left (x^2-16\right ) \log (x)}{64 \sqrt {x}}\right ) \]