<\body> |>>> Suppose we deposite > in a bank account, and the annual interest rate is . After years, <\enumerate> If the interest is compounded annualy, then the balance is <\equation*> u=u*. If the interest is compounded monthly, then the balance is <\equation*> u=u*|)> In general, if the interest is compounded times a year, then the balance is <\equation*> u=u*|)>. Taking the limit as \>, then for any fixed , we have <\equation*> u=lim\>|)>=lim\>|)>>>*r*t>=e. But we can obtain the same result by using a differential equation: <\equation> =u*r>. We can solve it later and obtain the same result. <\definition> A (ODE) about a function > is an equation involving > and its derivatives. Any ODE can be written in the abstract form <\equation> F,u,\,u>|)>=0. For example, Eq. (1) can be written as <\equation*> F|)>=r*u-u=0. In Eq. (2), is the of the equation, i.e. the order is the highest order derivative of in the equation.\ So Eq. (1) is an ODE of order 1.\ If is linear in terms of ,\,u>>, then the equation is called . Otherwise it's called . The general form of a linear ODE is <\equation*> a*u>+a*u>+\+a*u+a=0. So Eq. (1) is linear. Examples of nonlinear equations: <\equation*> u-u=5,u*u+5x=e,u=sin u A solution of an ODE <\equation*> F,u,\,u>|)>=0 is a function > satisfying the equation, i.e. <\equation*> F,\,\,\,\>|)>=0. <\example> Can you give solutions of <\equation*> u=2*u possible solution: >, in fact, > is a solution for any constant . <\example> Can you give solutions of <\equation*> u+4*u=0 A possible solution is , another solution is =cos*2t.> In fact, any function <\equation*> u=a*sin 2*t+b*cos 2*t \; \; \; \; \; \; \; \; > <\initial> <\collection> <\references> <\collection> > > > <\auxiliary> <\collection> <\associate|toc> |1Motivation |.>>>>|> > |2Classification of ODE |.>>>>|> > |3Solutions of an ODE |.>>>>|> >