Obtain just a particular solution (the general solution can always be obtained easily by adding an a

Taylor Haines

Taylor Haines

Answered question

2022-02-22

Obtain just a particular solution (the general solution can alwaysbe obtained easily by adding an arbitrary multiple of a solution of the associated homogeneous equation). In these exercises, the forcing function is an elementary sinusoidal function. If the forcing function is cos5t, then a particular solution of the form A cos5t will not work. However, a linear combination of cos5t and sin5t will work. Thus, the form for xp is Acos5t+Bsin5t, where the constants A and B are determined by substituting the assumed form of the particular solution into the differential equation for x. In Exercises 47–52, the forcing function has several different terms. Use a form for xp consisting of the sum of the forms for each term.
dxdt+x=2e3t+sint

Answer & Explanation

mastifo5h

mastifo5h

Beginner2022-02-23Added 6 answers

Given differential equation is Non-homogenous differential equation. Solution of this give differential equation is given by sum of Complementary function and particular integral that is (CF+PI).
CF is calculated by solving left hand side of the differential equation.
Given differential equation is
dxdt+x=2e3t+sint
First write auxiliary equation corresponding to given differential equation and the solve to get C.F.
D+1=0
D=1
C.F.=aet
Now find the PI to get solution of given differential equation.
PI=1(D+1)(2e3t+sint)=1D+1(2e3t)+1D+1sint=24e3t+1×(D1)(D+1)(D1)sint=12e3t+D1D21sint=12e3t+D111sint=12e3t+costsint2
Solution of the given diff is given by sum of CF and PI
Therefore, solution is
x(t)=aet+12e3tcostsint2

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