Internal problem ID [2431]
Book: Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition,
2015
Section: Chapter 11, Series Solutions to Linear Differential Equations. Exercises for 11.5. page
771
Problem number: (a).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {x^{2} y^{\prime \prime }-\left (-x^{2}+x \right ) y^{\prime }+\left (x^{3}+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: 63
Order:=6; dsolve(x^2*diff(y(x),x$2)-(x-x^2)*diff(y(x),x)+(1+x^3)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (\ln \relax (x ) c_{2}+c_{1}\right ) \left (1-x +\frac {1}{2} x^{2}-\frac {5}{18} x^{3}+\frac {19}{144} x^{4}-\frac {167}{3600} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (x -\frac {3}{4} x^{2}+\frac {41}{108} x^{3}-\frac {89}{432} x^{4}+\frac {2281}{27000} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{2}\right ) x \]
✓ Solution by Mathematica
Time used: 0.004 (sec). Leaf size: 114
AsymptoticDSolveValue[x^2*y''[x]-(x-x^2)*y'[x]+(1+x^3)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 x \left (-\frac {167 x^5}{3600}+\frac {19 x^4}{144}-\frac {5 x^3}{18}+\frac {x^2}{2}-x+1\right )+c_2 \left (x \left (\frac {2281 x^5}{27000}-\frac {89 x^4}{432}+\frac {41 x^3}{108}-\frac {3 x^2}{4}+x\right )+x \left (-\frac {167 x^5}{3600}+\frac {19 x^4}{144}-\frac {5 x^3}{18}+\frac {x^2}{2}-x+1\right ) \log (x)\right ) \]