81.11.24 problem 15-23

Internal problem ID [21648]
Book : The Differential Equations Problem Solver. VOL. I. M. Fogiel director. REA, NY. 1978. ISBN 78-63609
Section : Chapter 15. Method of undetermined coefficients. Page 337.
Problem number : 15-23
Date solved : Thursday, October 02, 2025 at 07:59:26 PM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }-2 y^{\prime }-3 y&=2 \,{\mathrm e}^{x}-10 \sin \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=2 \\ y^{\prime }\left (0\right )&=4 \\ \end{align*}

✓ Maple. Time used: 0.016 (sec). Leaf size: 29

ode:=diff(diff(y(x),x),x)-2*diff(y(x),x)-3*y(x) = 2*exp(x)-10*sin(x); 
ic:=[y(0) = 2, D(y)(0) = 4]; 
dsolve([ode,op(ic)],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.077 (sec). Leaf size: 37

ode=D[y[x],{x,2}]-2*D[y[x],x]-3*y[x]==2*Exp[x]-10*Sin[x]; 
ic={y[0]==2,Derivative[1][y][0] ==4}; 
DSolve[{ode,ic},y[x],x,IncludeSingularSolutions->True]
 

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

✓ Sympy. Time used: 0.152 (sec). Leaf size: 29

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-3*y(x) - 2*exp(x) + 10*sin(x) - 2*Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): 2, Subs(Derivative(y(x), x), x, 0): 4} 
dsolve(ode,func=y(x),ics=ics)
 

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