Internal problem ID [9084]
Book: Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section: Chapter 3, linear third order
Problem number: 1505.
ODE order: 3.
ODE degree: 1.
CAS Maple gives this as type [[_3rd_order, _with_linear_symmetries]]
Solve \begin {gather*} \boxed {2 x \left (x -1\right ) y^{\prime \prime \prime }+3 \left (2 x -1\right ) y^{\prime \prime }+\left (2 x a +b \right ) y^{\prime }+a y=0} \end {gather*}
✓ Solution by Maple
Time used: 1.125 (sec). Leaf size: 79
dsolve(2*x*(x-1)*diff(diff(diff(y(x),x),x),x)+3*(2*x-1)*diff(diff(y(x),x),x)+(2*a*x+b)*diff(y(x),x)+a*y(x)=0,y(x), singsol=all)
\[ y \relax (x ) = c_{1} \MathieuC \left (-\frac {a}{2}-\frac {b}{2}+1, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )^{2}+c_{2} \MathieuS \left (-\frac {a}{2}-\frac {b}{2}+1, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )^{2}+c_{3} \MathieuC \left (-\frac {a}{2}-\frac {b}{2}+1, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right ) \MathieuS \left (-\frac {a}{2}-\frac {b}{2}+1, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right ) \]
✓ Solution by Mathematica
Time used: 60.208 (sec). Leaf size: 115
DSolve[a*y[x] + (b + 2*a*x)*y'[x] + 3*(-1 + 2*x)*y''[x] + 2*(-1 + x)*x*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
\begin{align*} y(x)\to c_3 \text {MathieuC}\left [-\frac {a}{2}-\frac {b}{2}+1,\frac {a}{4},\text {ArcCos}\left (\sqrt {x}\right )\right ] \text {MathieuS}\left [-\frac {a}{2}-\frac {b}{2}+1,\frac {a}{4},\text {ArcCos}\left (\sqrt {x}\right )\right ]+c_1 \text {MathieuC}\left [-\frac {a}{2}-\frac {b}{2}+1,\frac {a}{4},\text {ArcCos}\left (\sqrt {x}\right )\right ]^2+c_2 \text {MathieuS}\left [-\frac {a}{2}-\frac {b}{2}+1,\frac {a}{4},\text {ArcCos}\left (\sqrt {x}\right )\right ]^2 \\ \end{align*}