0001 - CHAPTER 1 FIRST-ORDER DIFFERENTIAL EQUATIONS SECTION...

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Unformatted text preview: CHAPTER 1 FIRST-ORDER DIFFERENTIAL EQUATIONS SECTION 1.1 DIFFERENTIAL EQUATIONS AND MATHEMATICAL MODELING The main purpose of Section 1.1 is simply to introduce the basic notation and terminology of differential equations, and to show the student what is meant by a solution of a differential equation. Also, the use of differential equations in the mathematical modeling of real-world phenomena is outlined. Problems 142 are routine verifications by direct substitution of the suggested solutions into the given differential equations. We include here just some typical examples of such verifications. 3. If yl =cos 2x and y2 =sin 2x, then y,’:—2sin 2x and y; =2cos 2.7: so )7: : —4c052x = —4y, and y; = —4sin2x = —4y2. Thus yf+4yl = 0 and y§+4y2 = 0. 4. If yl 2:33" and y2 :34", then y, =3e3’ and y: 2—3841 so . Hence and y:+4y;+4y2 = (—4e’zx+4xc72x)+4(e‘2‘—2xe_2")+4[xe'h) = 0. 8. If y. =cosx—c052x and y2 =sinx—cos2x, then y::—sinx+25in2x. yfz—cosx+4cos 2x, and y; :cosx-t—Zsin 2x, y;=—sinx+4cos2x. Hence Section 1.1 ...
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