Internal problem ID [2475]
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: 11.
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 }+\frac {3 y^{\prime } x}{2}-\frac {\left (x +1\right ) y}{2}=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: 47
Order:=6; dsolve(x^2*diff(y(x),x$2)+3/2*x*diff(y(x),x)-1/2*(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {3}{2}} \left (1+\frac {1}{5} x +\frac {1}{70} x^{2}+\frac {1}{1890} x^{3}+\frac {1}{83160} x^{4}+\frac {1}{5405400} x^{5}+\mathrm {O}\left (x^{6}\right )\right )+c_{1} \left (1-x -\frac {1}{2} x^{2}-\frac {1}{18} x^{3}-\frac {1}{360} x^{4}-\frac {1}{12600} x^{5}+\mathrm {O}\left (x^{6}\right )\right )}{x} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 86
AsymptoticDSolveValue[x^2*y''[x]+3/2*x*y'[x]-1/2*(1+x)*y[x]==0,y[x],{x,0,5}]
\[ y(x)\to c_1 \sqrt {x} \left (\frac {x^5}{5405400}+\frac {x^4}{83160}+\frac {x^3}{1890}+\frac {x^2}{70}+\frac {x}{5}+1\right )+\frac {c_2 \left (-\frac {x^5}{12600}-\frac {x^4}{360}-\frac {x^3}{18}-\frac {x^2}{2}-x+1\right )}{x} \]