6.10.32 problem 32

Internal problem ID [1836]
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 : 32
Date solved : Tuesday, September 30, 2025 at 05:20:17 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} \left (x -1\right )^{2} y^{\prime \prime }-\left (x^{2}-1\right ) y^{\prime }+\left (x +1\right ) y&=\left (x -1\right )^{3} {\mathrm e}^{x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=4 \\ y^{\prime }\left (0\right )&=-6 \\ \end{align*}

✓ Maple. Time used: 0.025 (sec). Leaf size: 19

ode:=(x-1)^2*diff(diff(y(x),x),x)-(x^2-1)*diff(y(x),x)+(1+x)*y(x) = (x-1)^3*exp(x); 
ic:=[y(0) = 4, D(y)(0) = -6]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = {\mathrm e}^{x} x^{2}-{\mathrm e}^{x}-5 x +5 \]

✓ Mathematica. Time used: 0.027 (sec). Leaf size: 18

ode=(x-1)^2*D[y[x],{x,2}]-(x^2-1)*D[y[x],x]+(x+1)*y[x]==(x-1)^3*Exp[x]; 
ic={y[0]==4,Derivative[1][y][0] ==-6}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to (x-1) \left (e^x (x+1)-5\right ) \end{align*}

✗ Sympy

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-(x - 1)**3*exp(x) + (x - 1)**2*Derivative(y(x), (x, 2)) + (x + 1)*y(x) - (x**2 - 1)*Derivative(y(x), x),0) 
ics = {y(0): 4, Subs(Derivative(y(x), x), x, 0): -6} 
dsolve(ode,func=y(x),ics=ics)
 

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