6.10.31 problem 31

Internal problem ID [1835]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.7 Variation of Parameters. Page 262
Problem number : 31
Date solved : Tuesday, September 30, 2025 at 05:20:15 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

\begin{align*} y \left (0\right )&=3 \\ y^{\prime }\left (0\right )&=-6 \\ \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 32

ode:=(x-1)^2*diff(diff(y(x),x),x)-2*(x-1)*diff(y(x),x)+2*y(x) = (x-1)^2; 
ic:=[y(0) = 3, D(y)(0) = -6]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \left (-i \pi x +\ln \left (x -1\right ) x +i \pi -\ln \left (x -1\right )+2 x -3\right ) \left (x -1\right ) \]

✓ Mathematica. Time used: 0.019 (sec). Leaf size: 30

ode=(x-1)^2*D[y[x],{x,2}]-2*(x-1)*D[y[x],x]+2*y[x]==(x-1)^2; 
ic={y[0]==3,Derivative[1][y][0] ==-6}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to (x-1) (-i \pi (x-1)+2 x+(x-1) \log (x-1)-3) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq((x - 1)**2*Derivative(y(x), (x, 2)) - (x - 1)**2 - (2*x - 2)*Derivative(y(x), x) + 2*y(x),0) 
ics = {y(0): 3, Subs(Derivative(y(x), x), x, 0): -6} 
dsolve(ode,func=y(x),ics=ics)
 

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