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math54 - quiz 9 - solns

math54 - quiz 9 - solns - Math 54 quiz 9 Solutions 1 Find...

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Math 54 quiz 9 Solutions 1. Find the general solution to the differential equation y + 2 y + ky = 0 , where (a) k = 0, (b) k = 1, and (c) k = 2. (a) The roots of the characteristic equation are 0, and -2, so the solution is y ( x ) = c 1 + c 2 e - 2 x . (b) The root - 1 is a repeated root, so the solution is y ( x ) = c 1 e - x + c 2 xe - x . (c) The (complex) roots are - 1 ± i so the solution is y ( x ) = e - x ( c 1 sin( x ) + c 2 cos( x )) . 2. Suppose that y 1 satisfies the differential equation x 2 y + sin( x ) y = f ( x ) , and that y 2 satisfies the differential equation x 2 y + sin( x ) y = g ( x ) . Show that c 1 y 1 + c 2 y 2 satisfies x 2 y + sin( x ) y = c 1 f ( x ) + c 2 g ( x ) . In order to show that c 1 y 1 + c 2 y 2 satisfies the differential equation, we need to plug it in to the left hand side and see what we get: substituting c 1 y 1 + c 2 y 2 in for y , we obtain x 2 ( c 1 y 1 + c 2 y 2 ) +sin x ( c 1 y 1 + c 2 y 2 ) = c 1 x 2 y 1 + c 2 x 2 y 2 + c 1 sin( x ) y 1 + c 2 sin( x ) y 2 = c 1 ( x 2 y 1 + sin( x ) y 1 ) + and recalling that y 1 and y 2 satisfy the differential equations given above, this is c 1 f ( x ) + c 2 g ( x ) , as required. 1
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3. Find the general solution to the differential equation y - y - 2
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