Nonhomogeneous Differential Equations : Particular and General Solutions
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 = ...
Solution Summary
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.
(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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