60.4.35 problem 1491

Internal problem ID [11458]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 3, linear third order
Problem number : 1491
Date solved : Sunday, March 30, 2025 at 08:22:39 PM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.030 (sec). Leaf size: 88

ode:=x^2*diff(diff(diff(y(x),x),x),x)+3*x*diff(diff(y(x),x),x)+(4*a^2*x^(2*a)+1-4*nu^2*a^2)*diff(y(x),x) = 4*a^3*x^(2*a-1)*y(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 \operatorname {hypergeom}\left (\left [-\frac {1}{2}\right ], \left [-\nu +1, \nu +1\right ], -x^{2 a}\right )+c_2 \,x^{-2 a \nu } \operatorname {hypergeom}\left (\left [-\frac {1}{2}-\nu \right ], \left [1-2 \nu , -\nu +1\right ], -x^{2 a}\right )+c_3 \,x^{2 a \nu } \operatorname {hypergeom}\left (\left [-\frac {1}{2}+\nu \right ], \left [2 \nu +1, \nu +1\right ], -x^{2 a}\right ) \]

✓ Mathematica. Time used: 0.031 (sec). Leaf size: 102

ode=(1 - 4*a^2*nu^2 + 4*a^2*x^(2*a))*D[y[x],x] + 3*x*D[y[x],{x,2}] + x^2*Derivative[3][y][x] == 4*a^3*x^(-1 + 2*a)*y[x]; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to c_2 \left (x^{2 a}\right )^{-\nu } \, _1F_2\left (-\nu -\frac {1}{2};1-2 \nu ,1-\nu ;-x^{2 a}\right )+c_3 \left (x^{2 a}\right )^{\nu } \, _1F_2\left (\nu -\frac {1}{2};\nu +1,2 \nu +1;-x^{2 a}\right )+c_1 \, _1F_2\left (-\frac {1}{2};1-\nu ,\nu +1;-x^{2 a}\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
nu = symbols("nu") 
y = Function("y") 
ode = Eq(-4*a**3*x**(2*a - 1)*y(x) + x**2*Derivative(y(x), (x, 3)) + 3*x*Derivative(y(x), (x, 2)) + (-4*a**2*nu**2 + 4*a**2*x**(2*a) + 1)*Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (4*a**3*x**(2*a - 1)*y(x) - x**2*Derivative(y(x), (x, 3)) - 3*x*Derivative(y(x), (x, 2)))/(-4*a**2*nu**2 + 4*a**2*x**(2*a) + 1) cannot be solved by the factorable group method