84.28.4 problem 16.8

Internal problem ID [22281]
Book : Schaums outline series. Differential Equations By Richard Bronson. 1973. McGraw-Hill Inc. ISBN 0-07-008009-7
Section : Chapter 16. Initial-value problems. Supplementary problems
Problem number : 16.8
Date solved : Thursday, October 02, 2025 at 08:37:02 PM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

\begin{align*} y \left (1\right )&=2 \\ y^{\prime }\left (1\right )&=1 \\ \end{align*}

✓ Maple. Time used: 0.029 (sec). Leaf size: 39

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

\[ y = {\mathrm e}^{1-x}+\frac {{\mathrm e}^{4-x}}{12}+{\mathrm e}^{-2+2 x}-\frac {{\mathrm e}^{1+2 x}}{3}+\frac {{\mathrm e}^{3 x}}{4} \]

✓ Mathematica. Time used: 0.013 (sec). Leaf size: 50

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

\begin{align*} y(x)&\to \frac {1}{12} e^{-x-2} \left (12 e^{3 x}-4 e^{3 x+3}+3 e^{4 x+2}+12 e^3+e^6\right ) \end{align*}

✓ Sympy. Time used: 0.132 (sec). Leaf size: 36

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

\[ y{\left (x \right )} = \frac {e^{3 x}}{4} + \frac {\left (3 - e^{3}\right ) e^{2 x}}{3 e^{2}} + \left (e + \frac {e^{4}}{12}\right ) e^{- x} \]