60.5.33 problem 1570

Internal problem ID [11530]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1570
Date solved : Sunday, March 30, 2025 at 08:24:16 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime \prime \prime }+\left (6-4 a -4 c \right ) x^{3} y^{\prime \prime \prime }+\left (-2 \nu ^{2} c^{2}+2 a^{2}+4 \left (a +c -1\right )^{2}+4 \left (a -1\right ) \left (c -1\right )-1\right ) x^{2} y^{\prime \prime }+\left (2 \nu ^{2} c^{2}-2 a^{2}-\left (2 a -1\right ) \left (2 c -1\right )\right ) \left (2 a +2 c -1\right ) x y^{\prime }+\left (\left (-\nu ^{2} c^{2}+a^{2}\right ) \left (-\nu ^{2} c^{2}+a^{2}+4 a c +4 c^{2}\right )-b^{4} c^{4} x^{4 c}\right ) y&=0 \end{align*}

✓ Maple. Time used: 0.091 (sec). Leaf size: 47

ode:=x^4*diff(diff(diff(diff(y(x),x),x),x),x)+(6-4*a-4*c)*x^3*diff(diff(diff(y(x),x),x),x)+(-2*nu^2*c^2+2*a^2+4*(a+c-1)^2+4*(a-1)*(c-1)-1)*x^2*diff(diff(y(x),x),x)+(2*nu^2*c^2-2*a^2-(2*a-1)*(2*c-1))*(2*a+2*c-1)*x*diff(y(x),x)+((-c^2*nu^2+a^2)*(-c^2*nu^2+a^2+4*a*c+4*c^2)-b^4*c^4*x^(4*c))*y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \left (\operatorname {BesselJ}\left (\nu , i b \,x^{c}\right ) c_3 +\operatorname {BesselY}\left (\nu , b \,x^{c}\right ) c_2 +\operatorname {BesselY}\left (\nu , i b \,x^{c}\right ) c_4 +\operatorname {BesselJ}\left (\nu , b \,x^{c}\right ) c_1 \right ) x^{a} \]

✓ Mathematica. Time used: 0.067 (sec). Leaf size: 213

ode=((a^2 - c^2*\[Nu]^2)*(a^2 + 4*a*c + 4*c^2 - c^2*\[Nu]^2) - b^4*c^4*x^(4*c))*y[x] + (-1 + 2*a + 2*c)*(-2*a^2 - (-1 + 2*a)*(-1 + 2*c) + 2*c^2*\[Nu]^2)*x*D[y[x],x] + (-1 + 2*a^2 + 4*(-1 + a)*(-1 + c) + 4*(-1 + a + c)^2 - 2*c^2*\[Nu]^2)*x^2*D[y[x],{x,2}] + (6 - 4*a - 4*c)*x^3*Derivative[3][y][x] + x^4*Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to b^{a/c} (-1)^{\frac {a-c \nu }{4 c}} 2^{-\frac {2 a}{c}-\nu -3} \left (x^{4 c}\right )^{\frac {a}{4 c}} \left (4^{\nu } (4 c_1 \operatorname {Gamma}(1-\nu )-i c_2 \operatorname {Gamma}(2-\nu )) \operatorname {BesselJ}\left (-\nu ,b \sqrt [4]{x^{4 c}}\right )+4^{\nu } (4 c_1 \operatorname {Gamma}(1-\nu )+i c_2 \operatorname {Gamma}(2-\nu )) \operatorname {BesselI}\left (-\nu ,b \sqrt [4]{x^{4 c}}\right )+i^{\nu } \left ((4 c_3 \operatorname {Gamma}(\nu +1)-i c_4 \operatorname {Gamma}(\nu +2)) \operatorname {BesselJ}\left (\nu ,b \sqrt [4]{x^{4 c}}\right )+(4 c_3 \operatorname {Gamma}(\nu +1)+i c_4 \operatorname {Gamma}(\nu +2)) \operatorname {BesselI}\left (\nu ,b \sqrt [4]{x^{4 c}}\right )\right )\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
nu = symbols("nu") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 4)) + x**3*(-4*a - 4*c + 6)*Derivative(y(x), (x, 3)) + x**2*(2*a**2 - 2*c**2*nu**2 + (4*a - 4)*(c - 1) + 4*(a + c - 1)**2 - 1)*Derivative(y(x), (x, 2)) + x*(2*a + 2*c - 1)*(-2*a**2 + 2*c**2*nu**2 - (2*a - 1)*(2*c - 1))*Derivative(y(x), x) + (-b**4*c**4*x**(4*c) + (a**2 - c**2*nu**2)*(a**2 + 4*a*c - c**2*nu**2 + 4*c**2))*y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE Derivative(y(x), x) - (a**4*y(x) + 4*a**3*c*y(x) - 2*a**2*c**2*nu**2*y(x) + 4*a**2*c**2*y(x) + 6*a**2*x**2*Derivative(y(x), (x, 2)) - 4*a*c**3*nu**2*y(x) + 12*a*c*x**2*Derivative(y(x), (x, 2)) - 4*a*x**3*Derivative(y(x), (x, 3)) - 12*a*x**2*Derivative(y(x), (x, 2)) - b**4*c**4*x**(4*c)*y(x) + c**4*nu**4*y(x) - 4*c**4*nu**2*y(x) - 2*c**2*nu**2*x**2*Derivative(y(x), (x, 2)) + 4*c**2*x**2*Derivative(y(x), (x, 2)) - 4*c*x**3*Derivative(y(x), (x, 3)) - 12*c*x**2*Derivative(y(x), (x, 2)) + x**4*Derivative(y(x), (x, 4)) + 6*x**3*Derivative(y(x), (x, 3)) + 7*x**2*Derivative(y(x), (x, 2)))/(x*(4*a**3 + 12*a**2*c - 6*a**2 - 4*a*c**2*nu**2 + 8*a*c**2 - 12*a*c + 4*a - 4*c**3*nu**2 + 2*c**2*nu**2 - 4*c**2 + 4*c - 1)) cannot be solved by the factorable group method