60.4.44 problem 1492

Internal problem ID [11495]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1492
Date solved : Tuesday, January 28, 2025 at 06:06:37 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x^{2} y^{\prime \prime \prime }-3 \left (x -m \right ) x y^{\prime \prime }+\left (2 x^{2}+4 \left (n -m \right ) x +m \left (2 m -1\right )\right ) y^{\prime }-2 n \left (2 x -2 m +1\right ) y&=0 \end{align*}

✓ Solution by Maple

Time used: 0.112 (sec). Leaf size: 39

dsolve(x^2*diff(diff(diff(y(x),x),x),x)-3*(x-m)*x*diff(diff(y(x),x),x)+(2*x^2+4*(n-m)*x+m*(2*m-1))*diff(y(x),x)-2*n*(2*x-2*m+1)*y(x)=0,y(x), singsol=all)
 

\[ y = c_{1} \operatorname {KummerM}\left (-n , m , x\right )^{2}+c_{2} \operatorname {KummerU}\left (-n , m , x\right )^{2}+c_3 \operatorname {KummerM}\left (-n , m , x\right ) \operatorname {KummerU}\left (-n , m , x\right ) \]

✓ Solution by Mathematica

Time used: 0.194 (sec). Leaf size: 43

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

\[ y(x)\to c_2 \operatorname {HypergeometricU}(-n,m,x) L_n^{m-1}(x)+c_1 \operatorname {HypergeometricU}(-n,m,x)^2+c_3 L_n^{m-1}(x){}^2 \]