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Elimination of Arbitrary Constants

Introduction to elimination of arbitrary constants

The Differential Equations which have only one independent variable are called Ordinary. In the above equations x is the independent variable and y is the dependent variable. The Equations which have two or more independent variables and partial differential coefficients with respect to them are called Partial.

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The degree of an equation is defined as the degree of the highest differential coefficient while the equation has been made rational and integral regarding the differential coefficients are required in this equation.

The Benefits of Elimination of Arbitrary Constants

It encourages the optimization of the whole process, not yet individual elements. Local optimization could lead to imbalance .It force a user listen the  recognition of unnecessary complexity: which involves steps to add little value
It recognizes the role of suppliers in a derived area and the importance of up-stream can be  prevention while   down-stream cannot be detection
They  promotes standardization which around best practice

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Elimination of Arbitrary Constants in Differential Equations

Example:

To Eliminate the arbitrary constants from the  given equation:

y = Ce^x + De^-x + E

solution:-

differentiating  both sides  y ' = a E1 e ax cos bx - E1b  e ax sin bx + E2ae ax sin bx + E2b e ax cos bx

=  a { E1 e ax cos bx + E2 e ax sin bx } - E1 b ea x sin bx + E2 b e ax cos bx

= ay  - E1 b ea x sin bx + E2 b e ax cos bx ------(1)

Also,                     y ' - ay  = b { -E1  ea x sin bx + E2 e ax cos bx}- E1  ea x sin bx + E2  e ax cos bx = (1/b) {y ' - ay} -----(2)

differentiating again ,  y " = a y ' - E1 ab e ax sin bx - E1 b2 e ax sin bx + E2 ab e ax cos bx - E2 b2 e ax sin bx

=  a y '  + ab { - E1  e ax sin bx + E2  e ax cos bx }  - b2 { E1  e ax sin bx +E2  e ax sin bx }

= a y ' + ab { ( 1/b)(  y ' - ay) } -  a2 y - b2 y

= 2a y ' - a2 y - b2 y

Arbitrary constants are eliminated from the equation: y = Ce^x + De^-x + E The solution is detailed and well presented as above.

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Tags: elimination of arbitrary constants in di
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