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Unformatted text preview: MA1506 LECTURE NOTES CHAPTER 1 DIFFERENTIAL EQUATIONS 1.1 Introduction A differential equation is an equation that contains one or more derivatives of a differen tiable function. [In this course we deal only with ordinary DEs, NOT partial DEs.] The order of a d.e. is the order of the equa tion’s highest order derivative; and a d.e. is linear if it can be put in the form a n y ( n ) ( x )+ a n 1 y ( n 1) ( x )+ ··· + a 1 y (1) ( x )+ a y ( x ) = F, 1 where a i , ≤ i ≤ n , and F are all functions of x . For example, y = 5 y and xy sin x = 0 are first order linear d.e.; ( y 000 ) 2 + ( y 00 ) 5 y = e x is third order, nonlinear. We observe that in general, a d.e. has many solutions, e.g. y = sin x + c , c an arbitrary constant, is a solution of y = cos x . Such solutions containing arbitrary constants are called general solution of a given d.e.. Any solution obtained from the general solution by giving specific values to the arbitrary constants is called a particular solution of that d.e. e.g. 2 y = sin x + 1 is a particular solution of y = cos x . Basically, differential equations are solved us ing integration, and it is clear that there will be as many integrations as the order of the DE. Therefore, THE GENERAL SOLUTION OF AN nthORDER DE WILL HAVE n ARBI TRARY CONSTANTS. 1.2 Separable equations A first order d.e. is separable if it can be writ ten in the form M ( x ) N ( y ) y = 0 or equiva lently, M ( x ) dx = N ( y ) dy . When we write the 3 d.e. in this form, we say that we have separated the variables , because everything involving x is on one side, and everything involving y is on the other. We can solve such a d.e. by integrating w.r.t. x : Z M ( x ) dx = Z N ( y ) dy + c. Example 1. Solve y = (1 + y 2 ) e x . Solution. We separate the variables to obtain e x dx = 1 1 + y 2 dy. 4 Integrating w.r.t. x gives e x = tan 1 y + c, or tan 1 y = e x c, or y = tan( e x c ) . Example 2. Experiments show that a ra dioactive substance decomposes at a rate pro portional to the amount present. Starting with 2 mg at certain time, say t = 0, what can be said about the amount available at a later time? Example 3. A copper ball is heated to 5 100 ◦ C. At t = 0 it is placed in water which is maintained at 30 ◦ C. At the end of 3 mins the temperature of the ball is reduced to 70 ◦ C. Find the time at which the temperature of the ball is 31 ◦ C. Physical information: Experiments show that the rate of change dT/dt of the temperature T of the ball w.r.t. time is proportional to the dif ference between T and the temp T of the sur rounding medium. Also, heat flows so rapidly in copper that at any time the temperature is practically the same at all points of the ball....
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 Spring '08
 McNines
 Differential Equations, Equations, Derivative

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