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In Problems 11–16, determine whether the differential equation can be written in the separation of variables form
13.
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Calculus for Business, Economics, Life Sciences, and Social Sciences (14th Edition)
- In Problems 1–18 solve each differential equation by variation of parameters.y''+3y'+2y=sin(e^x)arrow_forwardOne of the below is a condition of a linear differential equation. A. dependent variable and its derivative are of degree greater than one B. Term coefficients should be dependent on the unknown variable C. coefficients of a term does not depend upon dependent variable D. all derivative process should be in terms of the constant variablearrow_forwardD. Exponential shipting Theorem 3 -3x 1. (D'+ 4D* - 3D - 18) y = 24 x earrow_forward
- 23. By suitably renaming the constants and dependent variables in the equations and T'=k(T-Tm) G' = -XG+r discussed in Section 1.2 in connection with Newton's law of cooling and absorption of glucose in the body, we can write both as y' = -ay+b, (A) (B) (C) where a is a positive constant and b is an arbitrary constant. Thus, (A) is of the form (C) with y = T, a = k, and b = kT, and (B) is of the form (C) with y = G, a = λ, and b = r. We'll encounter equations of the form (C) in many other applications in Chapter 2. Choose a positive a and an arbitrary b. Construct a direction field and plot some integral curves for (C) in a rectangular region of the form {0≤t≤T, c≤y≤d} of the ty-plane. Vary T, c, and d until you discover a common property of all the solutions of (C). Repeat this experiment with various choices of a and b until you can state this property precisely in terms of a and b.arrow_forwardPlease help. 1) A mathematical (differential equation) representation for a metapopulation can be written as dp/dt = cp(1- p) - ep where p is the proportion of patches occupied at time t, c is the rate of patch colonization, and e is the rate of patch extinction. The term on the left hand side is the ‘rate of change in the proportion of patches occupied.’ The first term on the right hand side of the equation provides the effective gain in patch occupancy as a function of p due to the colonization process, and the term in parentheses represents the fraction of patches not currently occupied. The second right hand side term provides the effective loss of patch occupancy due to the (local) extinction process. a) Pick a positive value for c and draw a plot that shows the colonization process (first right hand side term) as a function of p. b) Pick a positive value for e and draw a plot that shows the extinction process (second right hand side term) as a function of p. c) Look at the term…arrow_forward4. dr 0, x(0) = 1, x'(0) ; (0) - 3, x(0) = -1arrow_forward
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- State the order of the given ordinary differential equation. (1 - x)y" – 9xy' + 5y = cos x Determine whether the equation is linear or nonlinear by matching it with (6) in Section 1.1. dy + a,(x)y = g(x) (6) + a + · . . + a, (X) dx" - 1 dx O linear nonlineararrow_forward6. If g(x) = x'+x²+ g"(2). find the value ofarrow_forward7. Choose the appropriate table for the differential equation dx = x =x-y X y ଶ୪ X y dx X y dx -201 -4 12 2 -1 -1 -201 -4 12 2 -1 1 -201 -4 12 -2 -1 -1 Cannot be found without solving the differential equationarrow_forward
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