notes3_1 - MA366 Sathaye Notes on Chapter 3 Sec 1-3 1...

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MA366 Sathaye Notes on Chapter 3 Sec. 1-3 1. Notation. We shall use the following notation which is more convenient than those in the book. For the derivative d dt ( y ) = y 0 we prefer the simpler notation D t ( y ) or simply D ( y ) if the independent variable is understood. Then the second derivative takes on a convenient form D 2 ( y ) and more generally D m ( y ) becomes an m -th derivative. Next, an expression of the form y 00 + ty 0 +3 y can now be conveniently written as D 2 ( y )+ tD ( y )+ 3 y = ( D 2 + tD + 3)( y ). We may name the expression D 2 + tD + 3 as f ( D ) a certain differential operator and then write f ( D )( y ) or just f ( D ) y if there is no confusion. We will learn how to manipulate these differential operators as algebraic expressions and solve the differential equations effectively. 2. We should note that 3 D ( y ) = 3 y 0 = D (3 y ). However tD ( y ) = ty 0 while D ( ty ) = ( ty ) 0 = ty 0 + y = ( tD + 1)( y ). Thus tD ( y ) 6 = D ( ty ). Thus, we should note that while D commutes with constants, it does not commute with functions and we need to respect product rule in computation. 3. Homogeneous Equations with Constant Coefficients. A second order differential equation may be thought of as y 00 = F ( t, y, y 0 ) where F is some function. This is linear if it can be written in the form y 00 + p ( t ) y 0 + q ( t ) y = g ( t ) . In our operator notation, this will be shortened to ( D 2 + p ( t ) D + q ( t ))( y ) = g ( t ) ( * ) . Our second order equation ( * ) is said to be homogeneous if its RHS g ( t ) is zero. Warning: this is a different notion of homogeneous and should be distinguished from our earlier homogeneous separable equation.
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