47.4.11 problem 59

Internal problem ID [7488]
Book : Ordinary differential equations and calculus of variations. Makarets and Reshetnyak. Wold Scientific. Singapore. 1995
Section : Chapter 2. Linear homogeneous equations. Section 2.2 problems. page 95
Problem number : 59
Date solved : Sunday, March 30, 2025 at 12:10:27 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} x^{4} y^{\prime \prime \prime \prime }-x^{2} y^{\prime \prime }+y&=0 \end{align*}

✓ Maple. Time used: 0.004 (sec). Leaf size: 36

ode:=x^4*diff(diff(diff(diff(y(x),x),x),x),x)-x^2*diff(diff(y(x),x),x)+y(x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = \moverset {4}{\munderset {\textit {\_a} =1}{\sum }}x^{\operatorname {RootOf}\left (\textit {\_Z}^{4}-6 \textit {\_Z}^{3}+10 \textit {\_Z}^{2}-5 \textit {\_Z} +1, \operatorname {index} =\textit {\_a} \right )} \textit {\_C}_{\textit {\_a}} \]

✓ Mathematica. Time used: 0.005 (sec). Leaf size: 130

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

\[ y(x)\to c_4 x^{\text {Root}\left [\text {$\#$1}^4-6 \text {$\#$1}^3+10 \text {$\#$1}^2-5 \text {$\#$1}+1\&,4\right ]}+c_3 x^{\text {Root}\left [\text {$\#$1}^4-6 \text {$\#$1}^3+10 \text {$\#$1}^2-5 \text {$\#$1}+1\&,3\right ]}+c_1 x^{\text {Root}\left [\text {$\#$1}^4-6 \text {$\#$1}^3+10 \text {$\#$1}^2-5 \text {$\#$1}+1\&,1\right ]}+c_2 x^{\text {Root}\left [\text {$\#$1}^4-6 \text {$\#$1}^3+10 \text {$\#$1}^2-5 \text {$\#$1}+1\&,2\right ]} \]

✓ Sympy. Time used: 0.176 (sec). Leaf size: 90

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

\[ y{\left (x \right )} = C_{1} x^{\operatorname {CRootOf} {\left (x^{4} - 6 x^{3} + 10 x^{2} - 5 x + 1, 0\right )}} + C_{2} x^{\operatorname {CRootOf} {\left (x^{4} - 6 x^{3} + 10 x^{2} - 5 x + 1, 1\right )}} + C_{3} x^{\operatorname {CRootOf} {\left (x^{4} - 6 x^{3} + 10 x^{2} - 5 x + 1, 2\right )}} + C_{4} x^{\operatorname {CRootOf} {\left (x^{4} - 6 x^{3} + 10 x^{2} - 5 x + 1, 3\right )}} \]