By N. Finizio
An analogous, subtle usual Differential Equations with smooth functions by way of Finizio and Lades is the spine of this article. as well as this are incorporated purposes, ideas and conception of partial distinction equations, distinction equations and Fourier research.
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Extra resources for An Introduction to Differential Equations: With Difference Equations, Fourier Series, and Partial Differential Equations
That is the case in Exercises 1, 2 and 5, and Example 2. 2 the solutions of the differential equation (1) exist throughout the interval I, where the functions a(x) and b(x) are continuous. 2, implies that the REMARK 4 IVP y' + a(x)y = b(x) y(xo) = yo, where x,, E I, has a unique solution which exists throughout the interval I. The reader can verify by direct substitution (see Exercise 18) that the unique solution of the IVP is given by the formula oa ( :)d: + J y(x) = yoe b(s)e - ' a(r)dr ds. (5) ElementaryMethods-First-Order Differential Equations 1 32 EXAMPLE 1 Find the general solution of the differential equation d dx + (tan x) y = sin x in the interval (0, rr/2).
Multiplying both sides by exp /2 2 J 81 + 3t dr/I - exp 13 J 27d+ t) we obtain l In (27 + t)J = (27 + t)"', exp [ 3 \ [y(t)(27 + t)2" l' = 15(27 + t)2" Integrating both sides with respect to t, we obtain y(t)(27+t)l3=15(27+t)s's5+c and so y(t) = 9(27 + t) + c(27 + t) Using the initial condition (13), we find 20 = 9(27) + c(27) and so c = - 2007. 6 lb. EXERCISES In Exercises 1 through 14, some of the differential equations are linear and some are not. Determine those that are linear and solve them.
Thus, if N(t) is the number of atoms of the substance present at any time t, then Archaeology, N(t) _ -kN(t), Paleontology, Geology, Arts (15) Radioactive Dating 24 1 Elementary Methods-First-Order Differential Equations where the positive constant k is called the decay constant of the substance. ] If N(0) = No (16) is the (initial) number of atoms of the substance at time t = 0, the solution of the IVP (15)-(16) is given by (17) N(t) = N°e-". If we know the values of k, N(t), and N°, then from Eq.
An Introduction to Differential Equations: With Difference Equations, Fourier Series, and Partial Differential Equations by N. Finizio