Internal problem ID [1302]
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: 8.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {18 x^{2} \left (x +1\right ) y^{\prime \prime }+3 x \left (x^{2}+11 x +5\right ) y^{\prime }-\left (-5 x^{2}-2 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.017 (sec). Leaf size: 45
Order:=6; dsolve(18*x^2*(1+x)*diff(y(x),x$2)+3*x*(5+11*x+x^2)*diff(y(x),x)-(1-2*x-5*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} \sqrt {x}\, \left (1-\frac {1}{3} x +\frac {2}{15} x^{2}-\frac {5}{63} x^{3}+\frac {23}{405} x^{4}-\frac {458}{10395} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1-\frac {1}{12} x^{2}+\frac {1}{18} x^{3}-\frac {11}{288} x^{4}+\frac {31}{1080} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x^{\frac {1}{6}}} \]
✓ Solution by Mathematica
Time used: 0.006 (sec). Leaf size: 85
AsymptoticDSolveValue[18*x^2*(1+x)*y''[x]+3*x*(5+11*x+x^2)*y'[x]-(1-2*x-5*x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt [3]{x} \left (-\frac {458 x^5}{10395}+\frac {23 x^4}{405}-\frac {5 x^3}{63}+\frac {2 x^2}{15}-\frac {x}{3}+1\right )+\frac {c_2 \left (\frac {31 x^5}{1080}-\frac {11 x^4}{288}+\frac {x^3}{18}-\frac {x^2}{12}+1\right )}{\sqrt [6]{x}} \]