6.9.33 problem 33

Internal problem ID [1789]
Book : Elementary differential equations with boundary value problems. William F. Trench. Brooks/Cole 2001
Section : Chapter 5 linear second order equations. Section 5.6 Reduction or order. Page 253
Problem number : 33
Date solved : Tuesday, September 30, 2025 at 05:19:20 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

Using reduction of order method given that one solution is

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

With initial conditions

\begin{align*} y \left (0\right )&=1 \\ y^{\prime }\left (0\right )&=-1 \\ \end{align*}

✓ Maple. Time used: 0.051 (sec). Leaf size: 25

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

\[ y = \frac {\left (x +1\right ) \left (2 \,{\mathrm e}^{x} x -3 \,{\mathrm e}^{x}+7 \,{\mathrm e}^{-x}\right )}{4} \]

✓ Mathematica. Time used: 20.788 (sec). Leaf size: 5749

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

Too large to display

✗ 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)) - (2*x + 2)*Derivative(y(x), x) - (x**2 + 2*x - 1)*y(x),0) 
ics = {y(0): 1, Subs(Derivative(y(x), x), x, 0): -1} 
dsolve(ode,func=y(x),ics=ics)
 

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