SpecialTransformation

SpecialTransformation - Special Transformation There are...

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Special Transformation There are certain equations that can be transformed into a more basic type using a suitable transformation. The equations have the form: (a 1 x + b 1 y + c 1 ) dx + (a 2 x + b 2 y + c 2 ) dy = 0 where a 1 , b 1 , c 1 , a 2 , b 2 , c 2 are constants. There are two different kind of transformations according to relationships among the constants. Case 1 : a 2 a 1 ! b 2 b 1 Solve the system a 1 h + b 1 k + c 1 = 0 a 2 h + b 2 k + c 2 = 0 because of the imposed condition the system has a unique solution (h,k). Then, the transformation: x = X + h y = Y + k will change the original equation into a homogeneous equation in the variable X and Y, (a 1 X + b 1 Y) dX + (a 2 X + b 2 Y) dY = 0 Case 2 : a 2 a 1 = b 1 b 2 = k The transformation z = a 1 x + b 1 y changes the original equation into a separable equation in the variables z and x. Examples : Solve the equations 1) (2x - 5x + 3) dx – (2x + 4y – 6) dy = 0 Since 2/2 4/-5, let’s solve the system 2h - 5k + 3 = 0
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SpecialTransformation - Special Transformation There are...

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