Internal problem ID [5302]
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: 1(iii).
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {\left (-x^{2}+1\right ) x^{2} y^{\prime \prime }+3 \left (x^{2}+x \right ) y^{\prime }+y=0} \end {gather*} With the expansion point for the power series method at \(x = 0\).
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 81
Order:=8; dsolve((1-x^2)*x^2*diff(y(x),x$2)+3*(x+x^2)*diff(y(x),x)+y(x)=0,y(x),type='series',x=0);
\[ y \relax (x ) = \frac {\left (1+3 x +\frac {1}{2} x^{2}-\frac {1}{6} x^{3}+\frac {1}{16} x^{4}-\frac {43}{1200} x^{5}+\frac {161}{7200} x^{6}-\frac {1837}{117600} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) \left (\ln \relax (x ) c_{2}+c_{1}\right )+\left (\left (-9\right ) x -\frac {7}{2} x^{2}+\frac {7}{9} x^{3}-\frac {25}{96} x^{4}+\frac {5141}{36000} x^{5}-\frac {2083}{24000} x^{6}+\frac {489941}{8232000} x^{7}+\mathrm {O}\left (x^{8}\right )\right ) c_{2}}{x} \]
✓ Solution by Mathematica
Time used: 0.002 (sec). Leaf size: 84
AsymptoticDSolveValue[(1-x^2)*y''[x]+3*(x+x^2)*y'[x]+y[x]==0,y[x],{x,0,7}]
\[ y(x)\to c_2 \left (\frac {53 x^7}{630}+\frac {5 x^6}{24}+\frac {2 x^5}{15}-\frac {x^4}{4}-\frac {2 x^3}{3}+x\right )+c_1 \left (-\frac {19 x^7}{420}-\frac {x^6}{144}+\frac {3 x^5}{20}+\frac {5 x^4}{24}-\frac {x^2}{2}+1\right ) \]