Internal problem ID [5306]
Book: An introduction to Ordinary Differential Equations. Earl A. Coddington. Dover. NY
1961
Section: Chapter 4. Linear equations with Regular Singular Points. Page 166
Problem number: 3(d).
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 }-2 x \left (1+x \right ) y^{\prime }+2 \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: 0.062 (sec). Leaf size: 53
Order:=8; dsolve(x^2*diff(y(x),x$2)-2*x*(x+1)*diff(y(x),x)+2*(x+1)*y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = c_{1} x^{2} \left (1+x +\frac {2}{3} x^{2}+\frac {1}{3} x^{3}+\frac {2}{15} x^{4}+\frac {2}{45} x^{5}+\frac {4}{315} x^{6}+\frac {1}{315} x^{7}+\mathrm {O}\left (x^{8}\right )\right )+c_{2} x \left (1+2 x +2 x^{2}+\frac {4}{3} x^{3}+\frac {2}{3} x^{4}+\frac {4}{15} x^{5}+\frac {4}{45} x^{6}+\frac {8}{315} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.134 (sec). Leaf size: 92
AsymptoticDSolveValue[x^2*y''[x]-2*x*(x+1)*y'[x]+2*(1+x)*y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_1 \left (\frac {4 x^7}{45}+\frac {4 x^6}{15}+\frac {2 x^5}{3}+\frac {4 x^4}{3}+2 x^3+2 x^2+x\right )+c_2 \left (\frac {4 x^8}{315}+\frac {2 x^7}{45}+\frac {2 x^6}{15}+\frac {x^5}{3}+\frac {2 x^4}{3}+x^3+x^2\right ) \]