Internal problem ID [1421]
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: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} \left (3 x +2\right ) y^{\prime \prime }+x \left (4+21 x \right ) y^{\prime }-\left (1-9 x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.015 (sec). Leaf size: 69
Order:=6; dsolve(2*x^2*(2+3*x)*diff(y(x),x$2)+x*(4+21*x)*diff(y(x),x)-(1-9*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x \left (1-\frac {9}{4} x +\frac {135}{32} x^{2}-\frac {945}{128} x^{3}+\frac {25515}{2048} x^{4}-\frac {168399}{8192} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (-\frac {3}{4} x +\frac {27}{16} x^{2}-\frac {405}{128} x^{3}+\frac {2835}{512} x^{4}-\frac {76545}{8192} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (1-\frac {15}{4} x +\frac {63}{8} x^{2}-\frac {3699}{256} x^{3}+\frac {25623}{1024} x^{4}-\frac {1375137}{32768} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.045 (sec). Leaf size: 108
AsymptoticDSolveValue[2*x^2*(2+3*x)*y''[x]+x*(4+21*x)*y'[x]-(1-9*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {25515 x^{9/2}}{2048}-\frac {945 x^{7/2}}{128}+\frac {135 x^{5/2}}{32}-\frac {9 x^{3/2}}{4}+\sqrt {x}\right )+c_1 \left (\frac {3}{512} \sqrt {x} \left (945 x^3-540 x^2+288 x-128\right ) \log (x)+\frac {8613 x^4-5076 x^3+2880 x^2-1536 x+1024}{1024 \sqrt {x}}\right ) \]