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Nonhomogeneous Differential Equations : Particular and General Solutions

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14. Consider the nonhomogeneous linear equation

dy/dt = λy + cos(2t)

To find its general solution, we add the general solution of the associated homogeneous equation and a particular solution
yp(t) of the nonhomogeneous equation. Briefly explain why it does not matter which solution of the nonhomogeneous equation we use for yp(t).

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Consider two different solutions of the nonhomogeneous equation, y_1(t) and y_2(t).
Both of them satisfy the same differential equation:

dy_1/dt = λy_1 + cos(2t) (1)

dy_2/dt = ...

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Nonhomogeneous Differential Equations, Particular and General Solutions are investigated. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

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(1) Find a general solution to the following non homogeneous differential equation:

y'' + 5y' + 6y = 6x^2 + 10x + 2 + 12 e^x; yp(x) = e^x + x^2

(2) Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equations

(a) y'' - y' + y = (e^t + t)^2
(b) y'' + y' + y = e^t + 7
(c) y'' - 2y' + 3y = cosh t

(3) Find general solution to the following differential equations:

y'' + 4y = sin q - cos q

y'' + y' - 12 y = e^t + e2t - 1

y'' - 4y' + 4y = t^2 e^t - e^(2t)

(4) Find a particular solution to the given higher order differential equation:

y^(4) - 5y'' + 4y = 10 cos t - 20 sin t

Please help working on these attached problems

section 4.5 #6,10,12,14,20,28,36,38
Thanks

(6) Find a general solution to the following non homogeneous differential equation:

y'' + 5y' + 6y = 6x^2 + 10x + 2 + 12 e^x; yp(x) = e^x + x^2

Decide whether the method of undetermined coefficients together with superposition can be applied to find a particular solution of the given equations

(a) y'' - y' + y = (e^t + t)^2
(b) y'' + y' + y = e^t + 7
(c) y'' - 2y' + 3y = cosh t

Find general solution to the following differential equations:

(20) y'' + 4y = sin q - cos q
(28) y'' + y' - 12 y = e^t + e2t - 1
(36) y'' - 4y' + 4y = t^2 e^t - e^(2t)

Find a particular solution to the given higher order differential equation:

(38) y^(4) - 5y'' + 4y = 10 cos t - 20 sin t

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