Internal problem ID [11061]
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: 226.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (-x^{2}+1\right )^{2} y^{\prime \prime }-2 x \left (-x^{2}+1\right ) y^{\prime }+\left (\nu \left (\nu +1\right ) \left (-x^{2}+1\right )-\mu ^{2}\right ) y=0} \]
✓ Solution by Maple
Time used: 0.016 (sec). Leaf size: 17
dsolve((1-x^2)^2*diff(y(x),x$2)-2*x*(1-x^2)*diff(y(x),x)+(nu*(nu+1)*(1-x^2)-mu^2)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {LegendreP}\left (\nu , \mu , x\right )+c_{2} \operatorname {LegendreQ}\left (\nu , \mu , x\right ) \]
✓ Solution by Mathematica
Time used: 0.046 (sec). Leaf size: 20
DSolve[(1-x^2)^2*y''[x]-2*x*(1-x^2)*y'[x]+(\[Nu]*(\[Nu]+1)*(1-x^2)-\[Mu]^2)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 P_{\nu }^{\mu }(x)+c_2 Q_{\nu }^{\mu }(x) \]