Internal problem ID [6249]
Book: Elementary differential equations. Rainville, Bedient, Bedient. Prentice Hall. NJ. 8th edition.
1997.
Section: CHAPTER 18. Power series solutions. 18.11 Many-Term Recurrence Relations. Exercises page
391
Problem number: 10.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x y^{\prime \prime }+\left (1-x \right ) y^{\prime }-\left (1+x \right ) y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.047 (sec). Leaf size: 52
Order:=8; dsolve(2*x*diff(y(x),x$2)+(1-x)*diff(y(x),x)-(1+x)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} \sqrt {x}\, \left (1+\frac {1}{2} x +\frac {9}{40} x^{2}+\frac {103}{1680} x^{3}+\frac {187}{13440} x^{4}+\frac {247}{98560} x^{5}+\frac {17861}{46126080} x^{6}+\frac {23767}{461260800} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} \left (1+x +\frac {1}{2} x^{2}+\frac {1}{6} x^{3}+\frac {1}{24} x^{4}+\frac {1}{120} x^{5}+\frac {1}{720} x^{6}+\frac {1}{5040} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.023 (sec). Leaf size: 75
AsymptoticDSolveValue[x*(x-2)^2*y''[x]-2*(x-2)*y'[x]+2*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (-\frac {x^7}{5376}-\frac {x^6}{1920}-\frac {x^5}{640}-\frac {x^4}{192}-\frac {x^3}{48}-\frac {x^2}{8}+\frac {x}{2}+\left (1-\frac {x}{2}\right ) \log (x)\right )+c_1 \left (1-\frac {x}{2}\right ) \]