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Unformatted text preview: First Order Nonlinear Equations The most general nonlinear first order ordinary differential equation we could imagine would be of the form F t , y t , y t 0. 1 In general we would have no hope of solving such an equation. A less general nonlinear equation would be one of the form y t F t , y t , 2 but even this more general equation is often too difficult to solve. We will consider then, equations of the form y t F y t . 3 Equation (3) is said to be an autonomous differential equation, meaning that the nonlinear function F does not depend explicitly on t. The equation (2) is nonautonomous because F does contain explicit t dependence. The equations, y t y t 2 and y t y t 2 t 2 , are examples of autonomous and nonautonomous equations, respectively. We will consider some examples of nonlinear first order equations first and then state some general principles that will make it clear why autonomous equations are easier to deal with than nonautonomous ones. We will first recall a few of the properties that we have observed about linear problems. We saw in several examples that solutions to linear problems tend to be smooth functions, even when the coefficients and forcing term are discontinuous. The only thing that seemed to lead to a ”blow up” or singularity in the solution (i.e., a point where the solution becomes undefined) was a singularity in a coefficient or forcing term. Thus, when there are no singularities in the inputs of the problem, there will be no singularities in the solution and the solution will satisfy the equation for all t. Another way to say this is that there are no spontaneous singularities in the solution to a linear ODE. Solution singularities can only result from input singularities. In addition, the general solution of a linear equation is a 1-parameter family of functions which satisfies the equation for every choice of the parameter and which contains all possible solutions to the equation. That is, there are no solutions to the equation that can’t be written in the form of the general solution for some choice of the parameter. We havebe written in the form of the general solution for some choice of the parameter....
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- Summer '11