Internal problem ID [5704]
Book: Differential Equations: Theory, Technique, and Practice by George Simmons, Steven Krantz.
McGraw-Hill NY. 2007. 1st Edition.
Section: Chapter 4. Power Series Solutions and Special Functions. Section 4.4. REGULAR SINGULAR
POINTS. Page 175
Problem number: 4(d).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x^{2} y^{\prime \prime }+x 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: 0.018 (sec). Leaf size: 53
Order:=8; dsolve(2*x^2*diff(y(x),x$2)+x*diff(y(x),x)-(x+1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {c_{2} x^{\frac {3}{2}} \left (1+\frac {1}{5} x +\frac {1}{70} x^{2}+\frac {1}{1890} x^{3}+\frac {1}{83160} x^{4}+\frac {1}{5405400} x^{5}+\frac {1}{486486000} x^{6}+\frac {1}{57891834000} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{1} \left (1-x -\frac {1}{2} x^{2}-\frac {1}{18} x^{3}-\frac {1}{360} x^{4}-\frac {1}{12600} x^{5}-\frac {1}{680400} x^{6}-\frac {1}{52390800} x^{7}+\mathrm {O}\left (x^{8}\right )\right )}{\sqrt {x}} \]
✓ Solution by Mathematica
Time used: 0.003 (sec). Leaf size: 112
AsymptoticDSolveValue[2*x^2*y''[x]+x*y'[x]-(x+1)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 x \left (\frac {x^7}{57891834000}+\frac {x^6}{486486000}+\frac {x^5}{5405400}+\frac {x^4}{83160}+\frac {x^3}{1890}+\frac {x^2}{70}+\frac {x}{5}+1\right )+\frac {c_2 \left (-\frac {x^7}{52390800}-\frac {x^6}{680400}-\frac {x^5}{12600}-\frac {x^4}{360}-\frac {x^3}{18}-\frac {x^2}{2}-x+1\right )}{\sqrt {x}} \]