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notes 1-3 - y n = f t,y,y,y 00,y n-1 A solution of this ODE...

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SECTION 1.3 CLASSIFICATION OF DIFFERENTIAL EQUATIONS Ordinary versus partial, depending on the number of independent variables Order: the order of the highest derivative that appears Linear versus nonlinear, depending on whether anything is done to the unknown func- tion and its derivatives other than multiplying them by a function of the independent variable Suppose we have an ordinary differential equation (ODE) of order n . We always assume we can solve the ODE for the highest derivative, so we can write the equation as
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Unformatted text preview: y ( n ) = f ( t,y,y ,y 00 ,...,y ( n-1) ). A solution of this ODE on the interval α < t < β is a function φ ( t ) such that all the derivatives of φ ( t ) up to order n exist everywhere in the interval and satisfy φ ( n ) ( t ) = f ( t,φ ( t ) ,φ ( t ) ,...,φ ( n-1) ( t )) for every t in the interval. Example. For which values of r is y ( t ) = e rt a solution of y 00-9 y + 45 y = 0? HOMEWORK: SECTION 1.3...
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