Partial Differential Equations

There are 4 math questions in this task.
Q1: Using direct method & Laplace transform
Q2,3 : Sketch the forcing
Q4: derive two ODEs by separation of variables

[20 points] Consider the ordinary differential equation
1/’ + 63/ – 9y = 3e‘3′

for an unknown function y = y(t).

(a) Using the direct method (not the Laplace transform) find the general solution to this problem.

(b) Use the Laplace transform to solve the corresponding initial value problem

1/’ + 6y’ * 9y = 3e“”, 3/(0) = -1. y’(0) = 1-
[13 points] Solve the following initial value problem involving the Delta Dirac function 6 and the Heaviside
function H.
y” + 41/ + 5y = H(t – 10) + 6(t – 30), y(0) = 0, 3/(0) = 0.
Sketch the forcing and decide which of the graphs on the next page fits the solution best. Graph:
[13 points] Solve the following initial value problem with piecewise defined forcing.
2 , 3 5 t < 4
y +y= 0 th , . y(0)=0, y(0)=0-
o erunse
Sketch forcing and the solution into one plot.
[4 points] Consider the partial differential equation
62 6:: 8211
W 0′ E : W

for an unknown function u = u(:z:, t).

(a) If oz = a(t), that is, a is a function of t, but not of 2:. derive two oDEs by separation of variables.

(Do not try to solve the derived equations!)
(b) If oz = a(:z:). that is, a is a function of 2:, but not of t. derive two oDEs by separation of variables.
(Do not try to solve the derived equations!)

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