60.5.32 problem 1569

Internal problem ID [11529]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1569
Date solved : Thursday, March 13, 2025 at 08:53:38 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.203 (sec). Leaf size: 63

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

\[ y = x^{a} \left (\left (\operatorname {BesselJ}\left (\mu , b \,x^{c}\right ) c_{1} +\operatorname {BesselY}\left (\mu , b \,x^{c}\right ) c_3 \right ) \operatorname {BesselJ}\left (\nu , b \,x^{c}\right )+\operatorname {BesselY}\left (\nu , b \,x^{c}\right ) \left (c_4 \operatorname {BesselY}\left (\mu , b \,x^{c}\right )+\operatorname {BesselJ}\left (\mu , b \,x^{c}\right ) c_{2} \right )\right ) \]

✓ Mathematica. Time used: 0.065 (sec). Leaf size: 304

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

\[ y(x)\to b^{\frac {a-c (\mu +\nu )}{c}} \left (x^{2 c}\right )^{\frac {a-c (\mu +\nu )}{2 c}} \left (c_1 \, _2F_3\left (-\frac {\mu }{2}-\frac {\nu }{2}+\frac {1}{2},-\frac {\mu }{2}-\frac {\nu }{2}+1;1-\mu ,1-\nu ,-\mu -\nu +1;-b^2 x^{2 c}\right )+c_2 b^{2 \mu } \left (x^{2 c}\right )^{\mu } \, _2F_3\left (\frac {\mu }{2}-\frac {\nu }{2}+\frac {1}{2},\frac {\mu }{2}-\frac {\nu }{2}+1;\mu +1,1-\nu ,\mu -\nu +1;-b^2 x^{2 c}\right )+b^{2 \nu } \left (x^{2 c}\right )^{\nu } \left (c_3 \, _2F_3\left (-\frac {\mu }{2}+\frac {\nu }{2}+\frac {1}{2},-\frac {\mu }{2}+\frac {\nu }{2}+1;1-\mu ,\nu +1,-\mu +\nu +1;-b^2 x^{2 c}\right )+c_4 b^{2 \mu } \left (x^{2 c}\right )^{\mu } \, _2F_3\left (\frac {\mu }{2}+\frac {\nu }{2}+\frac {1}{2},\frac {\mu }{2}+\frac {\nu }{2}+1;\mu +1,\nu +1,\mu +\nu +1;-b^2 x^{2 c}\right )\right )\right ) \]

✗ Sympy

from sympy import * 
x = symbols("x") 
a = symbols("a") 
b = symbols("b") 
c = symbols("c") 
mu = symbols("mu") 
nu = symbols("nu") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 4)) + x**3*(6 - 4*a)*Derivative(y(x), (x, 3)) + x**2*(4*b**2*c**2*x**(2*c) - 2*c**2*(mu**2 + nu**2) + 6*(a - 1)**2 + 1)*Derivative(y(x), (x, 2)) + x*(b**2*c**2*x**(2*c)*(-8*a + 12*c + 4) + (2*a - 1)*(-2*a*(a - 1) + 2*c**2*(mu**2 + nu**2) - 1))*Derivative(y(x), x) + (b**2*c**2*x**(2*c)*(a - 2*c)*(4*a - 4*c) + (-a + c*mu - c*nu)*(-a + c*mu + c*nu)*(a + c*mu - c*nu)*(a + c*mu + c*nu))*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**2*b**2*c**2*x**(2*c)*y(x) + 2*a**2*c**2*mu**2*y(x) + 2*a**2*c**2*nu**2*y(x) - 6*a**2*x**2*Derivative(y(x), (x, 2)) + 12*a*b**2*c**3*x**(2*c)*y(x) + 4*a*x**3*Derivative(y(x), (x, 3)) + 12*a*x**2*Derivative(y(x), (x, 2)) - 8*b**2*c**4*x**(2*c)*y(x) - 4*b**2*c**2*x**(2*c + 2)*Derivative(y(x), (x, 2)) - c**4*mu**4*y(x) + 2*c**4*mu**2*nu**2*y(x) - c**4*nu**4*y(x) + 2*c**2*mu**2*x**2*Derivative(y(x), (x, 2)) + 2*c**2*nu**2*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 + 6*a**2 - 8*a*b**2*c**2*x**(2*c) + 4*a*c**2*mu**2 + 4*a*c**2*nu**2 - 4*a + 12*b**2*c**3*x**(2*c) + 4*b**2*c**2*x**(2*c) - 2*c**2*mu**2 - 2*c**2*nu**2 + 1)) cannot be solved by the factorable group method