56.36.1 problem Ex 1

Internal problem ID [14280]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter IX, Miscellaneous methods for solving equations of higher order than first. Article 60. Exact equation. Integrating factor. Page 139
Problem number : Ex 1
Date solved : Thursday, October 02, 2025 at 09:27:40 AM
CAS classification : [[_3rd_order, _missing_y]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 35

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

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

✓ Mathematica. Time used: 0.075 (sec). Leaf size: 45

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

\begin{align*} y(x)&\to x+\frac {1}{2} (c_1-c_2) (x+2) \cos (\log (x+2))+\frac {1}{2} (c_1+c_2) (x+2) \sin (\log (x+2))+c_3 \end{align*}

✗ Sympy

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

NotImplementedError : The given ODE x**2*Derivative(y(x), (x, 3)) + x*Derivative(y(x), (x, 2)) + 4*x*Derivative(y(x), (x, 3)) + Derivative(y(x), x) + 2*Derivative(y(x), (x, 2)) + 4*Derivative(y(x), (x, 3)) - 1 cannot be solved by the factorable group method