# Advanced Mathematics for Engineers by byrnetown70

VIEWS: 75 PAGES: 4

• pg 1
```									Advanced Mathematics for Engineers                          Questions

1. (a) Solve the ﬁrst order linear diﬀerential equation

dy
+ xy = x.
dx
[5]
(b) Give a condition for the diﬀerential equation

dy    M (x, y)
=−          ,
dx    N (x, y)

to be exact.                                                      [2]
k
Find the value k such that the integrating factor u(x) = (y + x)
makes the equation
dy            −y
=                       ,
dx   (y + x) ln(y + x) + y
exact.                                                        [8]
Hence ﬁnd the solution, you may leave your answer in the form
F (x, y) = C.
Hint:
y
ln(x + y) +         dy = y ln(y + x).
(y + x)
[5]

1
2. (a) Show that x = 0 is a regular singular point of the diﬀerential
equation

d2 y    1      dy      1
x2        +x   + 2x    + x−   y = 0.
dx2     2      dx      2
[3]
(b) Using the method of Frobenius we assume
∞
y=         an xn+r ,   a0 = 0.
n=0

Find the roots r1 , r2 of the indicial equation.                      [5]
(c) Show that the recurrence relationship for the unknown coeﬃcients
is given by

2n + 2r − 1
an = −an−1                             .
(n + r − 1)(n + r + 1/2)

Hence, ﬁnd the the coeﬃcients a1 , a2 , a3 for the two linearly inde-
pendent solutions.                                                    [12]

2
3. Consider the ﬁrst order initial value problem
dy
= f (x, y), y(x0 ) = y0 .
dx
The Euler method for this problem, using step size h, is given by

yk+1 = yk + hf (xk , yk ).

(a) By using Taylor’s remainder theorem for y(x) about x0 and as-
suming that y has a continuous second derivative, show that

|y(x1 ) − y1 | ≤ M h2 ,

where M is to be deﬁned.                                              [6]
(b) For
f (x, y) = xy 2 , y(0) = 5,
calculate the values yi , i = 0, 1, 2, 3, using step size h = 0.1. By
separating the variables ﬁnd the exact solution to this problem
and hence, compute the errors in the Euler approximation.             [7]
(c) Reduce the second order initial value problem

d2 y           dy               dy
2
= f (x, y, ), y(x0 ) = y0 , (x0 ) = z0 ,
dx             dx               dx
into a system of two ﬁrst order initial value problems. Hence,
write down a numerical scheme for approximating the solution to
a second order initial value problem using the Euler method.    [7]

3
4. Consider the problem

d2 y
−        + 4y = 1    on (0, 1),
dx2
y(0) = y(1) = 0.

(a) Find the exact solution to this problem.                            [6]
(b) Given equally spaced the nodes 0 = x0 < x1 < x2 < · · · < xn = 1,
we deﬁne
yi = y(xi ), i = 0, 1, 2, . . . , n − 1.
Deﬁne the centred ﬁnite diﬀerence approximation to the second
d2
derivative, dxy at the point x = xi , i = 1, 2, . . . , n − 1.
2

Using n = 4 obtain the linear system of equations
                                  
36 −16 0      y1       1
 −16 36 −16   y2  =  1               and y0 = y4 = 0.
                       

0 −16 36      y3       1
[7]
(c) Solve the linear system of equations.                               [7]

4

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