Internal problem ID [1358]
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: 6.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {9 x^{2} y^{\prime \prime }+3 x \left (-2 x^{2}+3 x +5\right ) y^{\prime }+\left (-14 x^{2}+12 x +1\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.031 (sec). Leaf size: 81
Order:=8; dsolve(9*x^2*diff(y(x),x$2)+3*x*(5+3*x-2*x^2)*diff(y(x),x)+(1+12*x-14*x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (c_{2} \ln \relax (x )+c_{1}\right ) \left (1-x +\frac {5}{6} x^{2}-\frac {1}{2} x^{3}+\frac {19}{72} x^{4}-\frac {43}{360} x^{5}+\frac {319}{6480} x^{6}-\frac {167}{9072} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (x -\frac {11}{12} x^{2}+\frac {25}{36} x^{3}-\frac {113}{288} x^{4}+\frac {4211}{21600} x^{5}-\frac {32773}{388800} x^{6}+\frac {126647}{3810240} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}}{x^{\frac {1}{3}}} \]
✓ Solution by Mathematica
Time used: 0.011 (sec). Leaf size: 168
AsymptoticDSolveValue[9*x^2*y''[x]+3*x*(5+3*x-2*x^2)*y'[x]+(1+12*x-14*x^2)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to \frac {c_1 \left (-\frac {167 x^7}{9072}+\frac {319 x^6}{6480}-\frac {43 x^5}{360}+\frac {19 x^4}{72}-\frac {x^3}{2}+\frac {5 x^2}{6}-x+1\right )}{\sqrt [3]{x}}+c_2 \left (\frac {\frac {126647 x^7}{3810240}-\frac {32773 x^6}{388800}+\frac {4211 x^5}{21600}-\frac {113 x^4}{288}+\frac {25 x^3}{36}-\frac {11 x^2}{12}+x}{\sqrt [3]{x}}+\frac {\left (-\frac {167 x^7}{9072}+\frac {319 x^6}{6480}-\frac {43 x^5}{360}+\frac {19 x^4}{72}-\frac {x^3}{2}+\frac {5 x^2}{6}-x+1\right ) \log (x)}{\sqrt [3]{x}}\right ) \]