75.19.10 problem 627

Internal problem ID [17055]
Book : A book of problems in ordinary differential equations. M.L. KRASNOV, A.L. KISELYOV, G.I. MARKARENKO. MIR, MOSCOW. 1983
Section : Chapter 2 (Higher order ODEs). Section 15.4 Nonhomogeneous linear equations with constant coefficients. The Euler equations. Exercises page 143
Problem number : 627
Date solved : Monday, March 31, 2025 at 03:39:24 PM
CAS classification : [[_3rd_order, _missing_y]]

\begin{align*} \left (2 x +1\right )^{2} y^{\prime \prime \prime }+2 \left (2 x +1\right ) y^{\prime \prime }+y^{\prime }&=0 \end{align*}

✓ Maple. Time used: 0.002 (sec). Leaf size: 50

ode:=(2*x+1)^2*diff(diff(diff(y(x),x),x),x)+2*(2*x+1)*diff(diff(y(x),x),x)+diff(y(x),x) = 0; 
dsolve(ode,y(x), singsol=all);
 

\[ y = c_1 +\frac {c_2 \left (2 x +1\right ) \sin \left (-\frac {\ln \left (2\right )}{2}+\frac {\ln \left (2 x +1\right )}{2}\right )}{2}+\frac {c_3 \left (2 x +1\right ) \cos \left (-\frac {\ln \left (2\right )}{2}+\frac {\ln \left (2 x +1\right )}{2}\right )}{2} \]

✓ Mathematica. Time used: 60.038 (sec). Leaf size: 46

ode=(2*x+1)^2*D[y[x],{x,3}]+2*(2*x+1)*D[y[x],{x,2}]+D[y[x],x]==0; 
ic={}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\[ y(x)\to \int _1^x\left (c_1 \cos \left (\frac {1}{2} \log (2 K[1]+1)\right )+c_2 \sin \left (\frac {1}{2} \log (2 K[1]+1)\right )\right )dK[1]+c_3 \]

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((2*x + 1)**2*Derivative(y(x), (x, 3)) + (4*x + 2)*Derivative(y(x), (x, 2)) + Derivative(y(x), x),0) 
ics = {} 
dsolve(ode,func=y(x),ics=ics)
 

NotImplementedError : The given ODE 4*x**2*Derivative(y(x), (x, 3)) + 4*x*Derivative(y(x), (x, 2)) + 4*x*Derivative(y(x), (x, 3)) + Derivative(y(x), x) + 2*Derivative(y(x), (x, 2)) + Derivative(y(x), (x, 3)) cannot be solved by the factorable group method