Using the method of undeterminedcoefficients to find the particular solution of the nonhomogeneous equation, find the solution of the following d.e. satisfying the given initial conditions
y"+4y=x^2+3e^x y(0)=0 y'(0)=2

1. Find a Particular Solution using undeterminedcoefficients, then find a general solution y³ + y" - 6y' = 3e²ⁿ.
2. Find a particular solution using the method of variation of parameters then find a general solution y" + 9y = cos(3x).
3. Solve the Initial Value Problem x" + 4x = 6sin(3t). x(0)=4, x'(0) = 0.
Ple

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 undeterminedcoefficients together with superposition

I am working on the differential equation
(dx^2)/(dt^2) + dx/dt + x =sin (ωt)
I have found the general solution of m^2+m+1=0 which is
x=e^(-1/2)t(Acos((sqrt3)/2)t+Bsin((sqrt3)/2)t
I am looking for a particular integral that satisfies the differential equation so as to obtain the general solution
I am finding great difficul

Please see the attached file for full description.
1. Find both the first and second order differentials (y' and y") for the following functions:
2. Use integrating factor to convert the following equation into "exact ODE" form and solve for y.
2xy' = (y -x) (y + x)/ y
3. Solve the differential equation, y is a functi

Differential Equation (IX): Formation of DifferentialEquations by Elimination
Eliminate the arbitrary constants from the equation: y = Ae^x + Be^2x + Ce^3x. Make sure to show all of the steps which are involved.

2. A spring with a 4-kg mass has natural length 1 m and is maintained stretched to a length of 1.3 m by a force of 24.3 N. If the spring is compressed to a length of 0.8 m and then released with zero velocity, find the position of the mass at any time t.
10. As in Exercise 9, consider a spring with mass m, spring constant k,

1. Write a short paragraph comparing and contrasting the method of undeterminedcoefficients and variation of parameters. How are they similar, how they are different? If you had your choice, which method would you use?
2. Consider the differential equation:
my"+cy'+ky=mg+sqrt(t)
Why would the method of undetermined

Using the method of undeterminedcoefficients, find the solution of the system:
X'=AX + B
that satisfies the initial condition:
X(0)=( 0
1
-1).
A and B are matrices defined in the attached Notepad file.
Note: When solving the homogeneous soln, exhibit a fundamental matrix psi(t) and al