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#### Find the critical points and phase portrait of the given autonomous first-order differential equation

###### Math

Find the critical points and phase portrait of the given autonomous first-order differential equation. Classify each critical point as asymptotically stable, unstable, or semi- stable. By hand, sketch typical solution curves in the regions in the xy-plane determined by the graphs of the equilibrium solutions. 5y3 - y dic 2 (10 pts) Suppose that a large mixing tank initially holds 300 gallons of water is which 50 pounds of salt have been dissolved. Another brine solution is pumped into the tank at a rate of 3 gal/min, and when the solution is well stirred, it is then pumped out at the rate of 3 gal/min. If the concentration of the solution entering is 2 lb/gal, determine a differential equation for the amount of salt A(t) in the tank at time t. Find the amount of salt in in grams in the tank at time t. 3 (10 pts) Use the method of variation of parameters to find the general solution of y' + y - by = 5e3+ 416 pts) A mass weighing 7 pounds is attached to a spring whose spring constant is 12 lb/ft. Find the equation of motion. (Use g = 32 ft/s2 for the acceleration due to gravity. Assume t is measured in seconds.) 5 Use the Laplace transform to solve the given initial-value problem. (click her for the Table of Laplace transforms e) a) (10 pts) y" + 4y = cos(t)u(t – 37), y(0) = 2, y'(0) = 0. b) (10 pts) y" +9y = f(t – 31) + (t – 57), y(0) = 1, y' (0) = 0. c) (7 pts) y' + 6y = et, y(0) = 2. Upload Choose a File x' +6y=e”, y(0) = 2: y=zze" 252-64 Steps + 6y = 27 Take Laplace transform of both sides of the equation L{v} + 6y} =[{7} Show Steps L{v} +6y): s[{y} – y(0) +62{y} L{e7t: Show Steps }: I 5-7 sl{y} - y(0) +62{v} = 1, 5-7 Plug in the initial conditions: y(0) = 2 sl{y} – 2 +62{y} 2 ==== Show Steps Isolate L{y}: [{y}= 2s - 13 (5-7)(3+6) Take the inverse Laplace transform 25 – 13 y=-. {62976446)} Show Steps [{{6297536)} 30%+42-60 y 1 77 13 - 25-61 13 54. f(t) F(s-a) 55. Ut -a) S 56. f(-a)U(1 - a) e-F(s) 57. g(1)U(t - a) a L{ \$(1 + a)) 58. f(t) ?"F(\$) - -f(0) - ... - f-) 59. tf(t) d" (-1)". ds" F(s) 60. S's(784 -- T)dt F(s)G(s) 61. 8(1) 1 62. 8(t - to)

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