62.35.2 problem Ex 2

Internal problem ID [12927]
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 : Monday, March 31, 2025 at 07:24:50 AM
CAS classification : [[_3rd_order, _with_linear_symmetries]]

\begin{align*} x y^{\prime \prime \prime }-y^{\prime \prime }-x y^{\prime }+y&=-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: 0.186 (sec). Leaf size: 170

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]
 

\[ y(x)\to x \left (\int _1^x-i \left (K[3]^2-1\right ) \left (y_{-2}(i K[3]) \int _1^{K[3]}j_{-2}(i K[1])dK[1]+j_{-2}(i K[3]) \int _1^{K[3]}-y_{-2}(i K[2])dK[2]\right )dK[3]+\int _1^x-y_{-2}(i K[2])dK[2] \left (\int _1^xi \left (K[5]^2-1\right ) j_{-2}(i K[5])dK[5]+c_3\right )+\int _1^xj_{-2}(i K[1])dK[1] \left (\int _1^xi \left (K[4]^2-1\right ) y_{-2}(i K[4])dK[4]+c_2\right )+c_1\right ) \]

✗ 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