56.35.2 problem Ex 2

Internal problem ID [14279]
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 59. Linear equations with particular integral known. Page 136
Problem number : Ex 2
Date solved : Friday, October 03, 2025 at 07:29:38 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} y-x y^{\prime }-y^{\prime \prime }+x y^{\prime \prime \prime }&=-x^{2}+1 \end{align*}

✓ Maple. Time used: 0.008 (sec). Leaf size: 22

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

\[ y = x^{2}+3+c_1 x +c_2 \,{\mathrm e}^{x}+c_3 \,{\mathrm e}^{-x} \]

✓ Mathematica. Time used: 0.109 (sec). Leaf size: 28

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

\begin{align*} y(x)&\to x^2+c_1 x-c_2 \cosh (x)+i c_3 \sinh (x)+3 \end{align*}

✗ Sympy

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

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