60.4.57 problem 1505

Internal problem ID [11508]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1505
Date solved : Tuesday, January 28, 2025 at 06:06:41 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} 2 x \left (x -1\right ) y^{\prime \prime \prime }+3 \left (2 x -1\right ) y^{\prime \prime }+\left (2 a x +b \right ) y^{\prime }+a y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.721 (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)
 

\[ y = c_{1} \operatorname {MathieuC}\left (-\frac {a}{2}-\frac {b}{2}+1, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )^{2}+c_{2} \operatorname {MathieuS}\left (-\frac {a}{2}-\frac {b}{2}+1, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right )^{2}+c_3 \operatorname {MathieuC}\left (-\frac {a}{2}-\frac {b}{2}+1, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right ) \operatorname {MathieuS}\left (-\frac {a}{2}-\frac {b}{2}+1, \frac {a}{4}, \arccos \left (\sqrt {x}\right )\right ) \]

✓ Solution by Mathematica

Time used: 60.212 (sec). Leaf size: 115

DSolve[a*y[x] + (b + 2*a*x)*D[y[x],x] + 3*(-1 + 2*x)*D[y[x],{x,2}] + 2*(-1 + x)*x*Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_3 \text {MathieuC}\left [-\frac {a}{2}-\frac {b}{2}+1,\frac {a}{4},\arccos \left (\sqrt {x}\right )\right ] \text {MathieuS}\left [-\frac {a}{2}-\frac {b}{2}+1,\frac {a}{4},\arccos \left (\sqrt {x}\right )\right ]+c_1 \text {MathieuC}\left [-\frac {a}{2}-\frac {b}{2}+1,\frac {a}{4},\arccos \left (\sqrt {x}\right )\right ]^2+c_2 \text {MathieuS}\left [-\frac {a}{2}-\frac {b}{2}+1,\frac {a}{4},\arccos \left (\sqrt {x}\right )\right ]^2 \]