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0013 - SECTION 1.4 SEPARABLE EQUATIONS AND APPLICATIONS Of...

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Unformatted text preview: SECTION 1.4 SEPARABLE EQUATIONS AND APPLICATIONS Of course it should be emphasized to students that the possibility of separating the variables is the first one you look for. The general concept of natural growth and decay is important for all differential equations students, but the particular applications in this section are optional. Torricelli's law in the form of Equation (24) in the text leads to some nice concrete examples and problems. 1. fl = “IQxdx; lny = _x2+c; y(x) = 6‘11” : Ce"? y 2. i); : ‘ngdx? *1 = -x2-C; 3U) = 21 y y x +C 3. [fl : Isinxdx; lny 2 —cosx+c; y(x) : em‘m = Ce'm“ y 4. [fl : idi; lny = 4ln(1+x)+lnC; y(x) 2 C(I+x)4 y l+x 5. JL= Ila; sin" y = 5+6 y(x) = sin(\/;+C) 1— 6 I ISde 2 y _ 2x‘”+2c y(x) _ (Wm) d 31'2 7. 3737 — I4xmdx, gym — 3 m+§C y(x) .- (2xm+C) 8. Icosydy = J2xdx; siny : x2+C; ny) z sin"(x2+C) 9. dy =ij de — —1—+; dx (partial fractions) l+x l—x lny=In(1+x)—1n(1—x)+1nc; y(x)= (3:3 “I Section 1.3 13 ...
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