ࡱ> *,)l` 0bjbj 8@ %" " " 6 %%%8&DZ&L6 8A2&B("d(d(d()))@@@@@@@jBhDH@Q" ))")))@d(d(/@,,,)^d(" d(@,)@,,r7T&" 8d(& N%*C8;@dA08AQ8E*E8E" 8L)),)))))@@,^)))8A))))6 6 6 $6 6 6 $6 6 6  Section 7.1: Modeling with Differential Equations Practice HW from Stewart Textbook (not to hand in) p. 503 # 1-7 odd Differential Equations Differential Equations are equations that contain an unknown function and one or more of its derivatives. Many mathematical models used to describe real-world problems rely on the use of differential equations (see examples on pp. 501-503). Most of the differential equations we will study in this chapter involve the first order derivative and are of the form  EMBED Equation.3  Our goal will be to find a function  EMBED Equation.3  that satisfies this equation. The following two examples illustrate how this can be done for a basic differential equation and introduce some basic terminology used when describing differential equations. Example 1: Find the general solution of the differential equation  EMBED Equation.3  Solution: % The general solution (or family of solutions) has the form  EMBED Equation.3 , where C is an arbitrary constant. When a particular value concerning the solution (known as an initial condition) of the form  EMBED Equation.3  (read as  EMBED Equation.3  when  EMBED Equation.3 ) is known, a particular solution, where a particular value of C is determined, can be found. The next example illustrates this. Example 2: Find the particular solution of the differential equation  EMBED Equation.3  Solution: % To check whether a given function is a solution of a differential equation, we find the necessary derivatives in the given equation and substitute. If the same quantity can be found on both sides of the equation, then the function is a solution. Example 3: Determine if the following functions are solutions to the differential equation  EMBED Equation.3 . a.  EMBED Equation.3  b.  EMBED Equation.3  Solution: % Example 4: Ve   13>TUfilorvwxy     : ; N O ˾}na}}RjH hKtCJUVaJjhKthKtEHUjH hKtCJUVaJjhKtUhKth-K h\ h1\ hc\ h$>K\huGh= hbh=h U5CJ aJ hZ5CJ aJ h+5CJ aJ hc5CJ aJ h=5CJ aJ 23fwxy    z { $a$gdKtgdKtgd=$a$gduG$a$gd=\000O P Q p   + b c v w x y z {    . V ⻮ʪҡҍҜzmhh hKt6jhKthKtEHUjKH hKtCJUVaJhU0 hn5 h(5 hU05 h-+@+B+D+F+H+J+L+N+P+R+gdq-L]Lgd^Ugdtn***<*>*@*B**********++++++0+2+D+x+++++++ʽ㹱yqllhh_ h!@5hq-hq-5jhrh_EHUj H h_CJUVaJjVhrh_EHUj>H h_CJUVaJjhrUhrjhrhrEHUjH hrCJUVaJjh2UUh2hq- hq-5 h"-5 h^U5 hH5'rify that  EMBED Equation.3  is a solution of the initial value problem  EMBED Equation.3  on the interval  EMBED Equation.3 . Solution: % Example 5: For what value of r does the function  EMBED Equation.3  satisfy the differential equation  EMBED Equation.3 ? Solution: For the function  EMBED Equation.3  to be a solution, we must, after computing the necessary derivative, obtain the same quantities on both sides of the equation after substitution. For  EMBED Equation.3 , we must compute, using the chain rule applied to the exponential function of base e,  EMBED Equation.3 . 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