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Unformatted text preview: a solution. For the linear rst order ODE x + a ( t ) x = b ( t ) we following existence-uniqueness theorem. Theorem 4.1 . If the function a ( t ) and b ( t ) are continuous on the open interval I containing the point t , then the initial value problem x + a ( t ) x = b ( t ) , x ( t ) = x has a unique solution x ( t ) on I, given by x ( t ) = e R a ( t ) dt Z b ( t ) e-R a ( t ) dt dt + C ! (8) with an appropriate value of C. So solving an initial value problem for linear rst order ODE is easy. One just apply the formula (8) and using the equation x ( t ) = x to nd C . Example 4.1 . Solve the initial value problem x +2 tx = 4 t 3 , x (0) = 1 . Solution First we apply the formula (8) x ( t ) = e R a ( t ) dt Z b ( t ) e-R a ( t ) dt dt + C ! ,...
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- Spring '11