Internal problem ID [1362]
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
II. Exercises 7.6. Page 374
Problem number: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {16 x^{2} y^{\prime \prime }+4 x \left (2 x^{2}+x +6\right ) y^{\prime }+\left (18 x^{2}+5 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.032 (sec). Leaf size: 81
Order:=8; dsolve(16*x^2*diff(y(x),x$2)+4*x*(6+x+2*x^2)*diff(y(x),x)+(1+5*x+18*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1-\frac {1}{4} x -\frac {7}{32} x^{2}+\frac {23}{384} x^{3}+\frac {145}{6144} x^{4}-\frac {881}{122880} x^{5}-\frac {4919}{2949120} x^{6}+\frac {47207}{82575360} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (\frac {1}{4} x +\frac {5}{64} x^{2}-\frac {157}{2304} x^{3}-\frac {841}{73728} x^{4}+\frac {65017}{7372800} x^{5}+\frac {50791}{58982400} x^{6}-\frac {953509}{1284505600} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}}{x^{\frac {1}{4}}} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 176
AsymptoticDSolveValue[16*x^2*y''[x]+4*x*(6+x+2*x^2)*y'[x]+(1+5*x+18*x^2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (\frac {47207 x^7}{82575360}-\frac {4919 x^6}{2949120}-\frac {881 x^5}{122880}+\frac {145 x^4}{6144}+\frac {23 x^3}{384}-\frac {7 x^2}{32}-\frac {x}{4}+1\right )}{\sqrt [4]{x}}+c_2 \left (\frac {-\frac {953509 x^7}{1284505600}+\frac {50791 x^6}{58982400}+\frac {65017 x^5}{7372800}-\frac {841 x^4}{73728}-\frac {157 x^3}{2304}+\frac {5 x^2}{64}+\frac {x}{4}}{\sqrt [4]{x}}+\frac {\left (\frac {47207 x^7}{82575360}-\frac {4919 x^6}{2949120}-\frac {881 x^5}{122880}+\frac {145 x^4}{6144}+\frac {23 x^3}{384}-\frac {7 x^2}{32}-\frac {x}{4}+1\right ) \log (x)}{\sqrt [4]{x}}\right ) \]