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# lecture_13 (dragged) - MA 36600 LECTURE NOTES MONDAY...

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MA 36600 LECTURE NOTES: MONDAY, FEBRUARY 16 Second Order Differential Equations Linear Equations. We briefly recall some facts about di ff erential equations which we have seen over the past few weeks. Recall that we use the notation y ( n ) = d n y dt n to denote the n th derivative. We say that an equation of the form F t, y, y (1) , . . . , y ( n ) = 0 is an n th order di ff erential equation . In particular, we can express the highest order derivative in terms of the lower order derivatives: d n y dt n = G t, y, y (1) , . . . , y ( n 1) for some function G ( t, y 1 , y 2 , . . . , y n ). When n = 2, we call an equation of the form d 2 y dt 2 = G t, y, dy dt a second order di ff erential equation . Recall that a first order di ff erential equation is said to be a linear equation if it is in the form dy dt = G ( t, y ) where G ( t, y ) = g ( t ) p ( t ) y for some functions p ( t ) and g ( t ). Similarly, we say that a second order di ff erential equation is a linear equation if it is in the form d 2 y dt 2 = G t, y, dy dt where G ( t, y 1 , y 2 ) = g ( t ) q ( t ) y p ( t ) dy dt for some functions p ( t ), q ( t
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