Internal problem ID [2476]
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. Additional problems. Section
11.7. page 788
Problem number: 12.
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 }-x \left (2-x \right ) y^{\prime }+\left (x^{2}+2\right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 65
Order:=6; dsolve(x^2*diff(y(x),x$2)-x*(2-x)*diff(y(x),x)+(2+x^2)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \left (\left (1-x +\frac {1}{3} x^{2}-\frac {1}{36} x^{3}-\frac {7}{720} x^{4}+\frac {31}{10800} x^{5}+\mathrm {O}\left (x^{6}\right )\right ) c_{1} x +c_{2} \left (\ln \relax (x ) \left (-x +x^{2}-\frac {1}{3} x^{3}+\frac {1}{36} x^{4}+\frac {7}{720} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+\left (1-x -\frac {1}{2} x^{2}+\frac {19}{36} x^{3}-\frac {53}{432} x^{4}-\frac {1}{675} x^{5}+\mathrm {O}\left (x^{6}\right )\right )\right )\right ) x \]
✓ Solution by Mathematica
Time used: 0.021 (sec). Leaf size: 85
AsymptoticDSolveValue[x^2*y''[x]-x*(2-x)*y'[x]+(2+x^2)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \left (\frac {1}{36} x^2 \left (x^3-12 x^2+36 x-36\right ) \log (x)-\frac {1}{432} x \left (65 x^4-372 x^3+648 x^2-432\right )\right )+c_2 \left (-\frac {7 x^6}{720}-\frac {x^5}{36}+\frac {x^4}{3}-x^3+x^2\right ) \]