89.23.11 problem 11

Internal problem ID [24815]
Book : A short course in Differential Equations. Earl D. Rainville. Second edition. 1958. Macmillan Publisher, NY. CAT 58-5010
Section : Chapter 10. Nonhomogeneous Equations: Operational methods. Miscellaneous Exercises at page 162
Problem number : 11
Date solved : Thursday, October 02, 2025 at 10:48:15 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

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

✓ Maple. Time used: 0.002 (sec). Leaf size: 23

ode:=diff(diff(y(x),x),x)-3*diff(y(x),x)+2*y(x) = (x-2)*exp(x); 
dsolve(ode,y(x), singsol=all);
 

\[ y = -\frac {\left (x^{2}-2 \,{\mathrm e}^{x} c_1 -2 x -2 c_2 \right ) {\mathrm e}^{x}}{2} \]

✓ Mathematica. Time used: 0.041 (sec). Leaf size: 33

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

\begin{align*} y(x)&\to \frac {1}{2} e^x \left (-x^2+2 x+2 \left (c_2 e^x+1+c_1\right )\right ) \end{align*}

✓ Sympy. Time used: 0.144 (sec). Leaf size: 19

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

\[ y{\left (x \right )} = \left (C_{1} + C_{2} e^{x} - \frac {x^{2}}{2} + x\right ) e^{x} \]