P2_130_2 - Review sheet for Math 415.01 Midterm II A second...

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Review sheet for Math 415.01 Midterm II A second order differential equation is called linear if it can be written in the form y 00 + p ( t ) y 0 + q ( t ) y = g ( t ) or P ( t ) y 00 + Q ( t ) y 0 + R ( t ) y = G ( t ). It is said to be homogeneous if the term g ( t ) (or the term G ( t )) is zero for all t . Otherwise the equation is called nonhomogeneous . If we have a homogeneous second-order differential equation with constant coefficients, such as ay 00 + by 0 + cy = 0, then we can assume that there are solutions of the form e rt for some constant r . If we plug this into the differential equation we get ( ar 2 + br + c ) e rt = 0, and since e rt is never zero, we get the characteristic equation ar 2 + br + c = 0, which we can solve for r . When the characteristic equation has two distinct real roots, r 1 and r 2 , the general solution of the differential equation can be written as y ( t ) = C 1 e r 1 t + C 2 e r 2 t . When the characteristic equation has a ± bi as roots 1 the general solution is y ( t ) = C 1 e at cos( bt ) + C 2 e at sin( bt ). When the characteristic equation has repeated root r 1 = r 2 (i.e. ar 2 + br + c is a perfect square) the general solution is y ( t ) = C 1 e r 1 t + C 2 te r 1 t . Theorem 3.2.1 Given the initial value problem y 00 + p ( t ) y 0 + q ( t ) y = g ( t ) , y ( t 0 ) = y 0 , y 0 ( t 0 ) = y 0 0 , where p ( t ), q ( t ), and g ( t ) are continuous on an open interval I that contains the point t 0 , there is exactly one solution y = φ ( t ) of this problem which exists throughout the interval I . Theorem 3.2.2 If y 1 and y 2 are two solutions of a linear differential equation, then for any constants C 1 and C 2 the linear combination C 1 y 1 + C 2 y 2 is also a solution.
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