23.5.175 problem 175

Internal problem ID [6784]
Book : Ordinary differential equations and their solutions. By George Moseley Murphy. 1960
Section : Part II. Chapter 5. THE EQUATION IS LINEAR AND OF ORDER GREATER THAN TWO, page 410
Problem number : 175
Date solved : Friday, October 03, 2025 at 02:09:52 AM
CAS classification : [[_high_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.009 (sec). Leaf size: 123

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

\[ y = c_1 \operatorname {hypergeom}\left (\left [k \right ], \left [a , b , c\right ], x\right )+c_2 \,x^{-a +1} \operatorname {hypergeom}\left (\left [1-a +k \right ], \left [2-a , 1-a +c , 1-a +b \right ], x\right )+c_3 \,x^{-b +1} \operatorname {hypergeom}\left (\left [1-b +k \right ], \left [2-b , 1-b +c , 1-b +a \right ], x\right )+c_4 \,x^{-c +1} \operatorname {hypergeom}\left (\left [1-c +k \right ], \left [2-c , 1-c +b , 1-c +a \right ], x\right ) \]

✓ Mathematica. Time used: 0.059 (sec). Leaf size: 143

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

\begin{align*} y(x)&\to x \left (-(-1)^{-a} c_2 x^{-a} \, _1F_3(-a+k+1;2-a,-a+b+1,-a+c+1;x)-(-1)^{-b} c_3 x^{-b} \, _1F_3(-b+k+1;2-b,a-b+1,-b+c+1;x)-(-1)^{-c} c_4 x^{-c} \, _1F_3(-c+k+1;2-c,a-c+1,b-c+1;x)\right )+c_1 \, _1F_3(k;a,b,c;x) \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE Derivative(y(x), x) - (-a*b*x*Derivative(y(x), (x, 2)) - a*c*x*D