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Differential Equations and Determinants

Question 1:
-----------

Quoting from the book:
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Example 2. Form a differential equation by eliminating the constants c1 and c2 from the equation

x 2x
y = c1*e + c2*e

Since there are two constants to eliminate, three equations are needed. The two extra equations are formed by differentiating the given equation twice, giving
2
dy x 2x d y x 2x
-- = c1*e + 2*c2*e and --- = c1*e + 4*c2*e
dx 2
dx

Eliminating, by determinants or otherwise, we have
2
d y dy
--- - 3 -- + 2y = 0
2 dx
dx
-----------------------------------------------------------------
Quotation ends.

I am perfectly comfortable (okay, at least reasonably comfortable ;-) with differential equation related parts of the problem; it's the "Eliminating, by determinants or otherwise" part that stymies me. How would I employ determinants (with which I have some familiarity) to obtain the above answer, and what are some of the other methods I could have used? (I never managed to obtain the final result, although I believe I did verify it.)

Question 2:
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In addition to having an interest, but no particular facility, in math, I am similarly predisposed towards (English) syntax (grammar). Perhaps this explains why my curiosity was piqued by the following question:

[In the interest of brevity, I shall edit more heavily this time.]

Quoting from the book:
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Example 1. Form a differential equation by eliminating the constant c from the equation x*y = c*sin(x).

...

...differentiating the given equation: x dy + y dx = c * cos(x) dx

Eliminating the c, we have the differential equation
dy
x -- + y = x*y*cot(x)
dx
-----------------------------------------------------------------
Quotation ends.

Once again, it is not the Diff Equation solving that I have a question with. It is the odd (to me) syntax of:

x dy + y dx = c * cos(x) dx

Of course I threw away my scrap paper, but I think I (eventually) solved the problem as follows:

x*y
x*y = c*sin(x) --> c = -----
sin(x)

Dx[x*y = c*sin(x)]

Dx[x*y] = Dx[c*sin(x)]

(by chain rule)
+- -+
| d(x) d(y) |
| ---- * y + x * ---- | = c * cos(x)
| dx dx |
+- -+

+- -+
| d(y) |
| y + x * ---- | = c * cos(x)
| dx |
+- -+

+- -+
| d(y) | x*y
| y + x * ---- | = ------ * cos(x)
| dx | sin(x)
+- -+

+- -+
| d(y) | cos(x)
| x * ---- + y | = ------ * x*y
| dx | sin(x)
+- -+

d(y)
x * ---- + y = cot(x) * x*y
dx
QED

You'll observe that at no time did I write something like:
"x dy + y dx = c * cos(x) dx"
(Or did I?)

Mr. Maurus was not a man to mince words; his entire "Differential Equations" book is less than 7in x 5in and numbers only about fifty pages. If he wrote "x dy + y dx = c * cos(x) dx", then I'm sure he knew what he was doing. Can you explain his terminology, or point me to a math book, web page, etc. that does?

Thanks in advance for your help!

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===== Q.1======= Response from OTA ================

Determinant method means solving a matrix or inverting a matrix..same as solving

simultaneous equations..but in this case, you can easily eliminate C1 and C2 as

follow:

I will use "^" for the exponent sign and use

an easier notation: call (dy/dx) = D; (d^2y/dx^2) = D^2

Let us say we want to find ...

Solution Summary

Differential equations and determinants are investigated.

$2.19