Internal problem ID [11001]
Book: Handbook of exact solutions for ordinary differential equations. By Polyanin and Zaitsev.
Second edition
Section: Chapter 2, Second-Order Differential Equations. section 2.1.2-5 Equation of form
\((a x^2+b x+c) y''+f(x)y'+g(x)y=0\)
Problem number: 167.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [[_2nd_order, _with_linear_symmetries]]
\[ \boxed {\left (-x^{2}+1\right ) y^{\prime \prime }-y^{\prime } x +\left (2 a \,x^{2}+b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.062 (sec). Leaf size: 27
dsolve((1-x^2)*diff(y(x),x$2)-x*diff(y(x),x)+(2*a*x^2+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {MathieuC}\left (a +b , -\frac {a}{2}, \arccos \left (x \right )\right )+c_{2} \operatorname {MathieuS}\left (a +b , -\frac {a}{2}, \arccos \left (x \right )\right ) \]
✓ Solution by Mathematica
Time used: 0.051 (sec). Leaf size: 34
DSolve[(1-x^2)*y''[x]-x*y'[x]+(2*a*x^2+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \text {MathieuC}\left [a+b,-\frac {a}{2},\arccos (x)\right ]+c_2 \text {MathieuS}\left [a+b,-\frac {a}{2},\arccos (x)\right ] \]