Internal problem ID [11050]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-7 Equation of form
\((a_4 x^4+a_3 x^3+a_2 x^2 x+a_1 x+a_0) y''+f(x)y'+g(x)y=0\)
Problem number: 225.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (x^{2}-1\right )^{2} y^{\prime \prime }+2 x \left (x^{2}-1\right ) y^{\prime }-\left (\nu \left (\nu +1\right ) \left (x^{2}-1\right )+n^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.063 (sec). Leaf size: 17
dsolve((x^2-1)^2*diff(y(x),x$2)+2*x*(x^2-1)*diff(y(x),x)-(nu*(nu+1)*(x^2-1)+n^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {LegendreP}\left (\nu , n , x\right )+c_{2} \operatorname {LegendreQ}\left (\nu , n , x\right ) \]
✓ Solution by Mathematica
Time used: 0.051 (sec). Leaf size: 20
DSolve[(x^2-1)^2*y''[x]+2*x*(x^2-1)*y'[x]-(\[Nu]*(\[Nu]+1)*(x^2-1)+n^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 P_{\nu }^n(x)+c_2 Q_{\nu }^n(x) \]