Question

8-11: Determine if the following problem is a separable or a linear differential equation or neither, then solve the first-order differential equation.If it is an initial value problem, find the exact solution. dy .dr a2y-ycos y(0) 3 8. 9. e(y-)da+ (1 + e*dy) = 0 10. (cos r)y (sin )y 1, y(T) = 1 shared via du dr 3yx-2a2- x with y(1) 0 11. a 12. In 1980 the population of alligators estimated to be 1500. In 2006 the population had grown to an estimated 6000. Using the Malthusian Law(Exponential) for population growth and estimate the alligator population on the Kennedy Space Center grounds in the year 2020 on the Kennedy Space Center ground was

Answer 1

Consider dy/dx = x^2 y - y cos x (1) with y (0) = 3 dy/dx = (x^2 - cos x) y separating variables 1/y dy = (x^2 - cos x) dx an integration integral 1/y dy = integral (x^2 - cos x) dx + ln c ln y = x^3/3 - sin x + ln c ln y - ln c = x^3/3 - sin x ln (y/c) = x^3/3 - sin (x) y/c = e^x^3/3 - sin (x) Hence y (x) = ce^x^3/3 - sin (x) using y (0) = 3 3 = Ce^0 (since sin (0) = 0) Hence c = 3 Thus y (x) = 3e^x^3/3 - sin (x) consider e^x (y - x) dx + (1 + e^x) dy = 0 (1) comparing with M dx + N dy = 0 M (x, y) = e^x (y - x) N = 1 + e^x Then Partialdifferential M/Partialdifferential y = e^x