60.4.20 problem 1468

Internal problem ID [11471]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1468
Date solved : Tuesday, January 28, 2025 at 06:06:32 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime \prime }-6 x y^{\prime \prime }+2 \left (4 x^{2}+2 a -1\right ) y^{\prime }-8 a x y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.156 (sec). Leaf size: 59

dsolve(diff(diff(diff(y(x),x),x),x)-6*x*diff(diff(y(x),x),x)+2*(4*x^2+2*a-1)*diff(y(x),x)-8*y(x)*a*x=0,y(x), singsol=all)
 

\[ y = x^{2} \left (\operatorname {KummerU}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )^{2} c_{2} +\operatorname {KummerU}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right ) \operatorname {KummerM}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right ) c_3 +\operatorname {KummerM}\left (\frac {1}{2}-\frac {a}{4}, \frac {3}{2}, x^{2}\right )^{2} c_{1} \right ) \]

✓ Solution by Mathematica

Time used: 0.044 (sec). Leaf size: 57

DSolve[-8*a*x*y[x] + 2*(-1 + 2*a + 4*x^2)*D[y[x],x] - 6*x*D[y[x],{x,2}] + Derivative[3][y][x] == 0,y[x],x,IncludeSingularSolutions -> True]
 

\[ y(x)\to c_2 \operatorname {HermiteH}\left (\frac {a}{2},x\right ) \operatorname {Hypergeometric1F1}\left (-\frac {a}{4},\frac {1}{2},x^2\right )+c_1 \operatorname {HermiteH}\left (\frac {a}{2},x\right )^2+c_3 \operatorname {Hypergeometric1F1}\left (-\frac {a}{4},\frac {1}{2},x^2\right )^2 \]