83.15.15 problem 5 (c)

Internal problem ID [22039]
Book : Differential Equations By Kaj L. Nielsen. Second edition 1966. Barnes and nobel. 66-28306
Section : Chapter XV. The Laplace Transform. Ex. XXIII at page 251
Problem number : 5 (c)
Date solved : Thursday, October 02, 2025 at 08:22:38 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }-3 y+8 \,{\mathrm e}^{-x}+3 x&=0 \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-{\frac {2}{3}} \\ y \left (1\right )&=2 \,{\mathrm e}^{-1}+\frac {1}{3} \\ \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 14

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)-3*y(x)+8*exp(-x)+3*x = 0; 
ic:=[y(0) = -2/3, y(1) = 2*exp(-1)+1/3]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

\[ y = -\frac {2}{3}+2 \,{\mathrm e}^{-x} x +x \]

✓ Mathematica. Time used: 0.024 (sec). Leaf size: 18

ode=D[y[x],{x,2}]-2*D[y[x],x]-3*y[x]+8*Exp[-x]+3*x==0; 
ic={y[0]==-2/3,y[1]==2*Exp[-1]+1/3}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

\begin{align*} y(x)&\to 2 e^{-x} x+x-\frac {2}{3} \end{align*}

✓ Sympy. Time used: 0.163 (sec). Leaf size: 14

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(3*x - 3*y(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)) + 8*exp(-x),0) 
ics = {y(0): -2/3, y(1): 1/3 + 2*exp(-1)} 
dsolve(ode,func=y(x),ics=ics)
 

\[ y{\left (x \right )} = x + 2 x e^{- x} - \frac {2}{3} \]