62.30.4 problem Ex 4

Internal problem ID [12897]
Book : An elementary treatise on differential equations by Abraham Cohen. DC heath publishers. 1906
Section : Chapter VIII, Linear differential equations of the second order. Article 53. Change of dependent variable. Page 125
Problem number : Ex 4
Date solved : Monday, March 31, 2025 at 07:23:54 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

✓ Maple. Time used: 0.005 (sec). Leaf size: 16

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

\[ y = c_2 x +{\mathrm e}^{x} c_1 +x^{2}+1 \]

✓ Mathematica. Time used: 0.232 (sec). Leaf size: 244

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

\[ y(x)\to \exp \left (\int _1^x\frac {K[1]-2}{2 (K[1]-1)}dK[1]-\frac {1}{2} \int _1^x-\frac {K[2]}{K[2]-1}dK[2]\right ) \left (\int _1^x\exp \left (\int _1^{K[4]}\frac {K[1]-2}{2 (K[1]-1)}dK[1]+\frac {1}{2} \int _1^{K[4]}-\frac {K[2]}{K[2]-1}dK[2]\right ) (K[4]-1) \int _1^{K[4]}\exp \left (-2 \int _1^{K[3]}\frac {K[1]-2}{2 (K[1]-1)}dK[1]\right )dK[3]dK[4]+\int _1^x\exp \left (-2 \int _1^{K[3]}\frac {K[1]-2}{2 (K[1]-1)}dK[1]\right )dK[3] \left (\int _1^x-\exp \left (\int _1^{K[5]}\frac {K[1]-2}{2 (K[1]-1)}dK[1]+\frac {1}{2} \int _1^{K[5]}-\frac {K[2]}{K[2]-1}dK[2]\right ) (K[5]-1)dK[5]+c_2\right )+c_1\right ) \]

✗ Sympy

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

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