Internal problem ID [8848]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 2, linear second order
Problem number: 1268.
ODE order: 2.
ODE degree: 1.
CAS Maple gives this as type [_Jacobi]
Solve \begin {gather*} \boxed {2 x \left (x -1\right ) y^{\prime \prime }+\left (2 x -1\right ) y^{\prime }+\left (a x +b \right ) y=0} \end {gather*}
✓ Solution by Maple
Time used: 0.043 (sec). Leaf size: 39
dsolve(2*x*(x-1)*diff(diff(y(x),x),x)+(2*x-1)*diff(y(x),x)+(a*x+b)*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \MathieuC \left (-a -2 b , \frac {a}{2}, \arccos \left (\sqrt {x}\right )\right )+c_{2} \MathieuS \left (-a -2 b , \frac {a}{2}, \arccos \left (\sqrt {x}\right )\right ) \]
✓ Solution by Mathematica
Time used: 0.078 (sec). Leaf size: 50
DSolve[(b + a*x)*y[x] + (-1 + 2*x)*y'[x] + 2*(-1 + x)*x*y''[x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_1 \text {MathieuC}\left [-a-2 b,\frac {a}{2},\text {ArcCos}\left (\sqrt {x}\right )\right ]+c_2 \text {MathieuS}\left [-a-2 b,\frac {a}{2},\text {ArcCos}\left (\sqrt {x}\right )\right ] \\ \end{align*}