Internal problem ID [1423]
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: 7.
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 }+4 y^{\prime } x -\left (9-x \right ) 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: 63
Order:=6; dsolve(4*x^2*diff(y(x),x$2)+4*x*diff(y(x),x)-(9-x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{3} \left (1-\frac {1}{16} x +\frac {1}{640} x^{2}-\frac {1}{46080} x^{3}+\frac {1}{5160960} x^{4}-\frac {1}{825753600} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (-\frac {1}{64} x^{3}+\frac {1}{1024} x^{4}-\frac {1}{40960} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (12+\frac {3}{2} x +\frac {3}{16} x^{2}-\frac {5}{4096} x^{4}+\frac {39}{819200} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )}{x^{\frac {3}{2}}} \]
✓ Solution by Mathematica
Time used: 0.024 (sec). Leaf size: 96
AsymptoticDSolveValue[4*x^2*y''[x]+4*x*y'[x]-(9-x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {x^{11/2}}{5160960}-\frac {x^{9/2}}{46080}+\frac {x^{7/2}}{640}-\frac {x^{5/2}}{16}+x^{3/2}\right )+c_1 \left (\frac {(x-16) x^{3/2} \log (x)}{12288}-\frac {19 x^4-64 x^3-2304 x^2-18432 x-147456}{147456 x^{3/2}}\right ) \]