Internal problem ID [1413]
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: Example 7.7.1 page 381.
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 (2+x \right ) y^{\prime \prime }-x \left (4-7 x \right ) y^{\prime }-\left (5-3 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: 65
Order:=6; dsolve(2*x^2*(2+x)*diff(y(x),x$2)-x*(4-7*x)*diff(y(x),x)-(5-3*x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{1} x^{3} \left (1-\frac {7}{4} x +\frac {63}{32} x^{2}-\frac {231}{128} x^{3}+\frac {3003}{2048} x^{4}-\frac {9009}{8192} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \left (\ln \relax (x ) \left (-\frac {45}{32} x^{3}+\frac {315}{128} x^{4}-\frac {2835}{1024} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (12+\frac {3}{2} x +\frac {9}{8} x^{2}-\frac {981}{64} x^{3}+\frac {6417}{256} x^{4}-\frac {28089}{1024} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.054 (sec). Leaf size: 98
AsymptoticDSolveValue[2*x^2*(2+x)*y''[x]-x*(4-7*x)*y'[x]-(5-3*x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_2 \left (\frac {3003 x^{13/2}}{2048}-\frac {231 x^{11/2}}{128}+\frac {63 x^{9/2}}{32}-\frac {7 x^{7/2}}{4}+x^{5/2}\right )+c_1 \left (\frac {15}{512} (7 x-4) x^{5/2} \log (x)+\frac {809 x^4-548 x^3+96 x^2+128 x+1024}{1024 \sqrt {x}}\right ) \]