20.7.12 problem Problem 36

Internal problem ID [3727]
Book : Differential equations and linear algebra, Stephen W. Goode and Scott A Annin. Fourth edition, 2015
Section : Chapter 8, Linear differential equations of order n. Section 8.3, The Method of Undetermined Coefficients. page 525
Problem number : Problem 36
Date solved : Sunday, March 30, 2025 at 02:06:38 AM
CAS classification : [[_2nd_order, _linear, _nonhomogeneous]]

\begin{align*} y^{\prime \prime }+y^{\prime }-2 y&=4 \cos \left (x \right )-2 \sin \left (x \right ) \end{align*}

With initial conditions

\begin{align*} y \left (0\right )&=-1\\ y^{\prime }\left (0\right )&=4 \end{align*}

✓ Maple. Time used: 0.031 (sec). Leaf size: 19

ode:=diff(diff(y(x),x),x)+diff(y(x),x)-2*y(x) = 4*cos(x)-2*sin(x); 
ic:=y(0) = -1, D(y)(0) = 4; 
dsolve([ode,ic],y(x), singsol=all);
 

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

✓ Mathematica. Time used: 0.018 (sec). Leaf size: 22

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

\[ y(x)\to -e^{-2 x}+e^x+\sin (x)-\cos (x) \]

✓ Sympy. Time used: 0.160 (sec). Leaf size: 19

from sympy import * 
x = symbols("x") 
y = Function("y") 
ode = Eq(-2*y(x) + 2*sin(x) - 4*cos(x) + Derivative(y(x), x) + Derivative(y(x), (x, 2)),0) 
ics = {y(0): -1, Subs(Derivative(y(x), x), x, 0): 4} 
dsolve(ode,func=y(x),ics=ics)
 

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