23.5.177 problem 177

Internal problem ID [6786]
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 : 177
Date solved : Tuesday, September 30, 2025 at 03:51:40 PM
CAS classification : [[_high_order, _with_linear_symmetries]]

\begin{align*} y+3 x y^{\prime }+9 x^{2} y^{\prime \prime }+6 x^{3} y^{\prime \prime \prime }+x^{4} y^{\prime \prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.001 (sec). Leaf size: 25

ode:=y(x)+3*x*diff(y(x),x)+9*x^2*diff(diff(y(x),x),x)+6*x^3*diff(diff(diff(y(x),x),x),x)+x^4*diff(diff(diff(diff(y(x),x),x),x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.003 (sec). Leaf size: 30

ode=y[x] + 3*x*D[y[x],x] + 9*x^2*D[y[x],{x,2}] + 6*x^3*D[y[x],{x,3}] + x^4*D[y[x],{x,4}] == 0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.199 (sec). Leaf size: 32

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(x**4*Derivative(y(x), (x, 4)) + 6*x**3*Derivative(y(x), (x, 3)) + 9*x**2*Derivative(y(x), (x, 2)) + 3*x*Derivative(y(x), x) + y(x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

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