9.11.29 problem 29

Internal problem ID [3139]
Book : Differential Equations by Alfred L. Nelson, Karl W. Folley, Max Coral. 3rd ed. DC heath. Boston. 1964
Section : Exercise 19, page 86
Problem number : 29
Date solved : Tuesday, September 30, 2025 at 06:27:28 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-3 y^{\prime }+2 y&=x \,{\mathrm e}^{-x} \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&={\frac {1}{9}} \\ y^{\prime }\left (0\right )&=0 \\ \end{align*}

✓ Maple. Time used: 0.027 (sec). Leaf size: 20

ode:=diff(diff(y(x),x),x)-3*diff(y(x),x)+2*y(x) = x*exp(-x); 
ic:=[y(0) = 1/9, D(y)(0) = 0]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = \frac {\left (6 x +5\right ) {\mathrm e}^{-x}}{36}-\frac {{\mathrm e}^{x}}{36} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 24

ode=D[y[x],{x,2}]-3*D[y[x],x]+2*y[x]==x*Exp[-x]; 
ic={y[0]==1/9,Derivative[1][y][0] ==0}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to -\frac {1}{36} e^{-x} \left (-6 x+e^{2 x}-5\right ) \end{align*}

✓ Sympy. Time used: 0.163 (sec). Leaf size: 17

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-x*exp(-x) + 2*y(x) - 3*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 1/9, Subs(Derivative(y(x), x), x, 0): 0} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = \frac {\left (6 x + 5\right ) e^{- x}}{36} - \frac {e^{x}}{36} \]