83.33.4 problem 4

Internal problem ID [19341]
Book : A Text book for differentional equations for postgraduate students by Ray and Chaturvedi. First edition, 1958. BHASKAR press. INDIA
Section : Chapter VII. Exact differential equations and certain particular forms of equations. Exercise VII (G) at page 115
Problem number : 4
Date solved : Monday, March 31, 2025 at 07:09:02 PM
CAS classification : [[_high_order, _missing_y]]

\begin{align*} x^{2} y^{\prime \prime \prime \prime }+a^{2} y^{\prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 43

ode:=x^2*diff(diff(diff(diff(y(x),x),x),x),x)+a^2*diff(diff(y(x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 +c_2 x +c_3 \,x^{\frac {5}{2}+\frac {\sqrt {-4 a^{2}+1}}{2}}+c_4 \,x^{\frac {5}{2}-\frac {\sqrt {-4 a^{2}+1}}{2}} \]

✓ Mathematica. Time used: 0.326 (sec). Leaf size: 100

ode=x^2*D[y[x],{x,4}]+a^2*D[y[x],{x,2}]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to -\frac {x^{\frac {1}{2} \left (5-\sqrt {\frac {1}{a^2}-4} a\right )} \left (\left (a^2+2 \sqrt {\frac {1}{a^2}-4} a-4\right ) c_2 x^{\sqrt {\frac {1}{a^2}-4} a}+\left (a^2-2 \sqrt {\frac {1}{a^2}-4} a-4\right ) c_1\right )}{a^4+8 a^2+12}+c_4 x+c_3 \]

✗ Sympy

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

NotImplementedError : CRootOf is not supported over ZZ[a]