Please help working on these attached problems
section 4.5 #6,10,12,14,20,28,36,38
(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
Following is the text part of the solution. Please see the attached file for complete solution. Equations, diagrams, graphs and special characters will not appear correctly here. Thank you for using Brainmass.
Corresponding Homogeneous equation:
y'' + 5 y' + 6y = 0
D^2 + 5D + 6 = 0
D = [-5 +/- sqrt(25 - 24)]/2 = -3 or -2
Hence the solution is a linear combination of e-3x and e-2x
yc = C1 e-3x + C2 e-2x
y(x) = ex + x2 + C1 e-3x + C2 e-2x
where C1 and C2 are arbitrary constants.
(10) (12) and (14) Answer is Yes to all three.
Let the given function be Ly = (e^t + t)^2 where L is the operator d^/dx^2 - d/dx + 1
Right hand side can be expanded to be e^(2t) + 2 t e^t + t^2
One can find a particular solution to each of the individual differential equation,
Ly = e^(2t)
Ly = 2 t e^t
Ly = t^2
Let the particular solutions of the above DE be, y1, y2 and y3.
Hence the ...
I have solved eight problems on differential equations. I have found general solutions and particular solutions to non homogeneous second order and higher order differential equations. I also have shown how to determine if the method of undetermined coefficients together with superposition can be utilized to solve given differential equations. This problem set contains a wide variety of problems encountered in upper level calculus course. This problem-solution set will be very useful in learning the solution process to differential equations and to prepare for upcoming examinations. Please download them for success in your calculus course.