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Differential equations

Homework 3

1.  Consider the following non-homogeneous differential equation:

with initial conditions

(a)  [20 points] Find the general solution of the problem by using the method of undetermined coefficients.

(b)  [20 points] Find the general solution of the problem by using the method of variation of parameters.

(c)  [20 points] Find the general solution of the problem by using the Laplace transform.

2.  Consider the beam bending equation

where F is the external force, EJ is the beam bending stiffness.  Solve the equation for each of the following boundary conditions:

(a)  [10 points] (Simply supported edges) w(0) = w′′ (0) = w(l) = w′′ (l) = 0;

(b)  [10 points] (Cantilevered edges) w(0) = w′ (0) = w(l) = w′ (l) = 0;

(c)  [10 points] (Simply supported-cantilevered edges) w(0) = w′′ (0) = w(l) = w′ (0);

(d)  [10 points] (Cantilevered-free edges) w(0) = w′ (0) = w′′ (l) = w′′′ (l) = 0.