2.11.17 problem 32

Internal problem ID [885]
Book : Differential equations and linear algebra, 3rd ed., Edwards and Penney
Section : Section 5.5, Nonhomogeneous equations and undetermined coefficients Page 351
Problem number : 32
Date solved : Tuesday, September 30, 2025 at 04:19:02 AM
CAS classification : [[_2nd_order, _with_linear_symmetries]]

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

With initial conditions

\begin{align*} y \left (0\right )&=0 \\ y^{\prime }\left (0\right )&=3 \\ \end{align*}

✓ Maple. Time used: 0.028 (sec). Leaf size: 21

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

\[ y = \frac {{\mathrm e}^{x}}{6}-\frac {8 \,{\mathrm e}^{-2 x}}{3}+\frac {5 \,{\mathrm e}^{-x}}{2} \]

✓ Mathematica. Time used: 0.014 (sec). Leaf size: 26

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

\begin{align*} y(x)&\to \frac {1}{6} e^{-2 x} \left (15 e^x+e^{3 x}-16\right ) \end{align*}

✓ Sympy. Time used: 0.125 (sec). Leaf size: 22

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

\[ y{\left (x \right )} = \frac {e^{x}}{6} + \frac {5 e^{- x}}{2} - \frac {8 e^{- 2 x}}{3} \]