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In Problems 11–16, determine whether the differential equation can be written in the separation of variables form
12.
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- In Problems 1–18 solve each differential equation by variation of parameters.y''+3y'+2y=sin(e^x)arrow_forwardD. Exponential shipting Theorem 3 -3x 1. (D'+ 4D* - 3D - 18) y = 24 x earrow_forward23. 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_forward
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- Is the question true or falsearrow_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_forward6. If g(x) = x'+x²+ g"(2). find the value ofarrow_forward
- 21. A differential equation may possess more than one family of solutions. (a) Plot different members of the families y = 1(x) = x² + c₁ and y = 2(x) = −x² + C₂. (b) Verify that y = 1(x) and y = 2(x) are two solutions of the nonlinear first-order differential equation (y)² = 4x². (c) Construct a piecewise-defined function that is a solution of the nonlinear DE in part (b) but is not a member of either family of solutions in part (a).arrow_forwardCalculate y" and y". 10e* X y(x) = y"(x) = y" (x) =arrow_forwardState 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_forward
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