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m2k_dfq_linfrd

m2k_dfq_linfrd - Linear First Order Dierential Equations...

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Linear First Order Differential Equations These equations take the general form dy dx = - p ( x ) y + g ( x ) , i . e ., dy dx + p ( x ) y = g ( x ) , where p ( x ) and g ( x ) are continuous (piecewise continuous is OK) func- tions on some interval a 0 x b 0 . We refer to these first order differ- ential equations as linear differential equations because the unknown y appears linearly, i.e., to the first power, with a known coefficient, p ( x ). The equation is homogeneous if g ( x ) 0, for given functions g ( x ) other than zero it is said to be inhomogeneous . Homogeneous Case: Equation: dy dx + p ( x ) y = 0. Properties: y ( x ) 0 is a solution; the trivial solution. (Clear.) If y ( x ) is any solution of the homogeneous equation then, for any constant c , the multiple c y ( x ) is also a solution of that equation. (Just substitute c y ( x ) into the equation.) If a solution y ( x ) is non-zero for some value, say x 1 , of x , then it is non-zero for all values of x . (Will be shown.) Method of Solution: In the homogenous case we have dy dx = - p ( x ) y . Assuming that we are looking for a non-zero solution, we can write 1 y ( x ) dy dx = - p ( x ) . Integrating both sides with respect to x , we have (natural logarithm) log | y ( x ) | = - Z x p ( s ) ds + ˆ c ≡ - P ( x ) + ˆ c, 1

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where ˆ c is an arbitrary constant and P ( x ) is an antiderivative, or in- definite integral, of p ( x ). Taking the exponential of both sides we have the general solution y ( x, c ) = e ˆ c exp {- Z x p ( s ) ds } ≡ c e - P ( x ) , where P ( x
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m2k_dfq_linfrd - Linear First Order Dierential Equations...

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