Find a particular solution to the differential equation.

avalg10o 2022-03-25 Answered
Find a particular solution to the differential equation.
yy6y=sint+3cost
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Answers (2)

Alannah Farmer
Answered 2022-03-26 Author has 11 answers
The given equation is a second order linear non homogeneous differential equation with constant coefficient.
The general solution for a(t)y+b(t)y+c(t)y=g(t)
The general solution of the given differential equation can be written as
y=yh+yp
yh is the solution to the homogeneous ODE a(t)y+b(t)y+c(t)y=0
and the particular solution, is any function that satisfies the non-homogeneous equation.
The complementary solution for the given equation is:
yy6y=sint+3cost
yy6y=0
(D2D6)y=0
λ2λ6=0
(λ3)(λ+2)=0
λ=3,λ=2
y=c1e3t+c2e2t
Two real and different root λ1 and λ2
The solution is of the form: y=c1eλ1t+c2eλ2t
So the solution is of the form:
yc=c1e3t+c2e2t
And the particular solution is given as:
(D2D6)y=sint+3cost
yp=sint+3costD2D6=1D2D6sint+3cost
=1D2D6sint+1D2D63cost
=1(1)D6sint+1(1)D63cost
=1D7sint+31D7cost
=D7(D+7)(D7)sint3D7(D7)(D+7)cost {Ratinalizing the denominator}
=D7D249sint3D7D249cost
=D7149sint3D7149cost
=D750sint+3D750cost
=cos(t)50750sint+3(sint)502150cost
=15sin(t)25cos(t)
Now, the solution of the given ODE is:
y=yp+yc
y(x)=c1e3t+c2e2t15sin(t)25cos(t)
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Jeffrey Jordon
Answered 2022-03-31 Author has 2262 answers

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