23.5.67 problem 67

Internal problem ID [6676]
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 : 67
Date solved : Friday, October 03, 2025 at 02:09:43 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} f \left (x \right ) y+y^{\prime }+f \left (x \right ) y^{\prime \prime }+y^{\prime \prime \prime }&=0 \end{align*}

✓ Maple. Time used: 0.007 (sec). Leaf size: 36

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

\[ y = {\mathrm e}^{i x} \left (\int {\mathrm e}^{-2 i x} \left (c_3 \int {\mathrm e}^{\int \left (-f \left (x \right )+i\right )d x}d x +c_2 \right )d x +c_1 \right ) \]

✓ Mathematica. Time used: 0.035 (sec). Leaf size: 84

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

\begin{align*} y(x)&\to c_3 e^{i x} \int _1^xe^{-2 i K[3]} \int _1^{K[3]}\exp \left (\int _1^{K[2]}(i-f(K[1]))dK[1]\right )dK[2]dK[3]+c_1 e^{i x}+\frac {1}{2} i c_2 e^{-i x} \end{align*}

✗ Sympy

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

NotImplementedError : solve: Cannot solve f(x)*y(x) + f(x)*Derivative(y(x), (x, 2)) + Derivative(y(x