Internal problem ID [1317]
Book: Elementary differential equations with boundary value problems. William F. Trench.
Brooks/Cole 2001
Section: Chapter 7 Series Solutions of Linear Second Equations. 7.5 THE METHOD OF FROBENIUS
I. Exercises 7.5. Page 358
Problem number: 28.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (8+x \right ) y^{\prime \prime }+x \left (3 x +2\right ) y^{\prime }+\left (x +1\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: 47
Order:=6; dsolve(x^2*(8+x)*diff(y(x),x$2)+x*(2+3*x)*diff(y(x),x)+(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{\frac {1}{4}} \left (1-\frac {25}{96} x +\frac {675}{14336} x^{2}-\frac {38025}{5046272} x^{3}+\frac {732615}{645922816} x^{4}-\frac {9230949}{56103010304} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{2} \sqrt {x}\, \left (1-\frac {9}{40} x +\frac {5}{128} x^{2}-\frac {245}{39936} x^{3}+\frac {6615}{7241728} x^{4}-\frac {7623}{57933824} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 90
AsymptoticDSolveValue[x^2*(8+x)*y''[x]+x*(2+3*x)*y'[x]+(1+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (-\frac {7623 x^5}{57933824}+\frac {6615 x^4}{7241728}-\frac {245 x^3}{39936}+\frac {5 x^2}{128}-\frac {9 x}{40}+1\right )+c_2 \sqrt [4]{x} \left (-\frac {9230949 x^5}{56103010304}+\frac {732615 x^4}{645922816}-\frac {38025 x^3}{5046272}+\frac {675 x^2}{14336}-\frac {25 x}{96}+1\right ) \]