Internal problem ID [11007]
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: 173.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
\[ \boxed {2 x \left (x -1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y=0} \]
✓ Solution by Maple
Time used: 0.078 (sec). Leaf size: 39
dsolve(2*x*(x-1)*diff(y(x),x$2)+(2*x-1)*diff(y(x),x)+(a*x+b)*y(x)=0,y(x), singsol=all)
\[ y \left (x \right ) = c_{1} \operatorname {MathieuC}\left (-2 b -a , \frac {a}{2}, \arccos \left (\sqrt {x}\right )\right )+c_{2} \operatorname {MathieuS}\left (-2 b -a , \frac {a}{2}, \arccos \left (\sqrt {x}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.258 (sec). Leaf size: 50
DSolve[2*x*(x-1)*y''[x]+(2*x-1)*y'[x]+(a*x+b)*y[x]==0,y[x],x,IncludeSingularSolutions -> True]
\[ y(x)\to c_1 \text {MathieuC}\left [-a-2 b,\frac {a}{2},\arccos \left (\sqrt {x}\right )\right ]+c_2 \text {MathieuS}\left [-a-2 b,\frac {a}{2},\arccos \left (\sqrt {x}\right )\right ] \]