Solving Difference Equation and Differential Equation

1. Complex Exponentials: Simply the following expression and give your answer both in polar and rectangular form.

o c=3ejπ/4+4e−jπ/2
2. Difference Equations: Solve the following difference equation using recursion by hand (for n=0 to n=4)
o y[n] + 0.5y[n-1] = 2x[n-1]; x[n] = δ[n], y[-1] = 0
3. Differential Equations: Solve the following problem for y(t).
o dy/dt + 2y(t) = 2x(t); x(t) = u(t), y(0) = −1

The solution gives detailed steps on solving difference equation, solving differential equation and simplifying complex exponential in polar and rectangular form.

A) Solve the following differentialequation by as many different methods as you can.
(See attachment for equation)
b) There is a type of differentialequation which will always be solvable by two different methods. What type of differentialequation is it and which other method can always be used to solve it?
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I already solved the homogeneous portion, and I need help solving the particular solution and of course combining the two to get the entire solution to the differentialequation. Not too difficult - see attachment. Please use equation editor if possible. Thank you.
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Given that:
dMS/dt = m(MN - MS) - pMS¬

1. Please see the attached file for the fully formatted problems.
a) Use separation of variables to solve
b) Solve the following exact differentialequation
c) By means of substitution y=vx solve the differentialequation
d) By means of the substitution for approipriate values of and , solve the d

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 differentialequation, y is a functi

57. Show that the substitution v = lny transforms the differentialequation
dy/dx + P(x)y = Q(x)(ylny) into the linear equation dv/dx + P(x) = Q(x)v(x)
58. Use the idea in Problem 57 to solve the equation
x (dy/dx) - 4x2y + 2ylny = 0
59. Solve the differentialequation dy/dx = (x-y-1)/(x+y+3) by finding h and k so t

DifferentialEquation (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.

Linear Partial DifferentialEquation (II)
Non- Homogeneous Linear Partial DifferentialEquation with Constant Coefficients
Problem: Find the solution of the equation (D2 - D'2 + D -

(1) Use Laplace Transforms to solve DifferentialEquation
y'' - 8y' + 20 y = t (e^t) , given that y(0) = 0 , y'(0) = 0
(2) Use Laplace Transforms to solve DifferentialEquation
y''' + 2y'' - y' - 2y = Sin 3t , given that y(0)=0 , y'(0)=0 ,y''(0)=0, y'''(0)=1
Note: To see the questions in their mathematic