54.5.16 problem 1553

Internal problem ID [12812]
Book : Differential Gleichungen, E. Kamke, 3rd ed. Chelsea Pub. NY, 1948
Section : Chapter 4, linear fourth order
Problem number : 1553
Date solved : Wednesday, October 01, 2025 at 02:22:14 AM
CAS classification : [[_high_order, _missing_y]]

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

✓ Maple. Time used: 0.003 (sec). Leaf size: 17

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

\[ y = \left (c_4 x +c_2 \right ) \ln \left (x \right )+c_3 x +c_1 \]

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 29

ode=2*D[y[x],{x,2}] + 4*x*Derivative[3][y][x] + x^2*Derivative[4][y][x] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to (c_4-2 c_2) x+(c_2 x-c_1) \log (x)+c_3 \end{align*}

✓ Sympy. Time used: 0.047 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = C_{1} + C_{2} x + C_{3} x \log {\left (x \right )} + C_{4} \log {\left (x \right )} \]