Find the particular integral of the following differential
equations.(Explain each step clearly)
(a) d2y/dx2 + y = (x + 1) sin x. show that
the answer is yp(x) = − 1/8 [ (2x2 + 4x − 1) cos x − (2x
+ 2) sin x ]
(Hint:In this case, we substitute sin αx or cos αx with
eiαx then use the shift operator. In the case of sin αx
we extract the imaginary part.)
Consider the equation: x2 y''-6y=0
A. Could you solve this ODE using Homogeneous Linear Equations
with Constant Coefficients? Explain.
B. Note that y1=x3 is a
solution of the ODE. Using reduction of order, find a solution
y2 such that { y1, y2} is linearly
independent.
C. Prove that{y1, y2} is linearly
independent.
D.What is the general solution?
Solve two different first order differential equations (one
linear and one non-linear) both analytically and numerically and
compare the results in tabular and graphical forms. Include at
least two different numerical solution techniques for each
differential equation analyzed.
Solve the following linear programming model graphically:
Max Z= 3x1 +4x2
Subject to: 2x1 + 4x2 <= 22
-x1 + 4x2 <= 10
4x1 – 2x2 <= 14 x1 – 3x2 <= 1
x1, x2, >=0
Clearly identify the feasible region, YOUR iso-profit line and
the optimal solution (that is, d.v. values and O.F. Value.