Internal problem ID [1357]
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: 5.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} \left (x^{2}+x +1\right ) y^{\prime \prime }-x \left (-2 x^{2}-4 x +1\right ) y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.018 (sec). Leaf size: 75
Order:=8; dsolve(x^2*(1+x+x^2)*diff(y(x),x$2)-x*(1-4*x-2*x^2)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-4 x +\frac {19}{2} x^{2}-\frac {49}{3} x^{3}+\frac {515}{24} x^{4}-\frac {319}{15} x^{5}+\frac {10093}{720} x^{6}-\frac {647}{360} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+\left (3 x -\frac {43}{4} x^{2}+\frac {208}{9} x^{3}-\frac {10379}{288} x^{4}+\frac {76321}{1800} x^{5}-\frac {172499}{4800} x^{6}+\frac {39091}{2400} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.013 (sec). Leaf size: 158
AsymptoticDSolveValue[x^2*(1+x+x^2)*y''[x]-x*(1-4*x-2*x^2)*y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 x \left (-\frac {647 x^7}{360}+\frac {10093 x^6}{720}-\frac {319 x^5}{15}+\frac {515 x^4}{24}-\frac {49 x^3}{3}+\frac {19 x^2}{2}-4 x+1\right )+c_2 \left (x \left (\frac {39091 x^7}{2400}-\frac {172499 x^6}{4800}+\frac {76321 x^5}{1800}-\frac {10379 x^4}{288}+\frac {208 x^3}{9}-\frac {43 x^2}{4}+3 x\right )+x \left (-\frac {647 x^7}{360}+\frac {10093 x^6}{720}-\frac {319 x^5}{15}+\frac {515 x^4}{24}-\frac {49 x^3}{3}+\frac {19 x^2}{2}-4 x+1\right ) \log (x)\right ) \]