please answer all the questions correctly.

1)

Which of the following is the solution to the differential equation y ' = (x + 1)() + 2) for y > -2, with the initial condition y(0) = 4? O +X 2 y = 60 -2 O x2 -+X 2 y = 40 Ova_y+2 X O -4X 2 y = 2e - 2What are all solutions to the differential equation y ' = 4x2 + x22? Oy'= 8x+ 2xy2 + C 2x y O y' 4x + + C 3 3 O y= =tan 3 I+c 2x3 O y = 2tan +C 3Which of the following is the particular solution to the differential equation = cos(x2), with the initial condition y dx = 3? 2 O y = 3+ [ cos(t?) dt y=3+/*_cos(t?) dt 12 sin(x?) 1 OV= +3- 2x Oy = sin(x2) + 2Let y: x) he the particular solution to the differential equation %- y' with the initial condition R3) = [Which ol the following gives an expression for x) and the doma'n for which the solution is valid? What is the general solution to the differential equation cosy(2In(x)- x)- dy 2 -3x27 dx X O y = arcsin In - - 3x +C X O y = -arcsin In - -3x2+c X O y = arcsin(In|2In(x) - x | + C) Oy= -arcsin(In 2In(x) - x=[) + CThe value of a laptop t years after it is purchased is given by the decreasing function V, where V(t) is measured in dollars. The rate of change of the laptop's value in dollars per year is proportional to the laptop's value. Which of the following differential equations, with the correct description of constant k, could be used to model the value of the laptop? Odv v -, where k is positive. Odv k dt "V' where k is negative. O dv = kv, where k is negative. O av = kt, where k is positive.The rate at which a particular drug leaves an individual's bloodstream is proportional to the amount of this drug that is in the bloodstream. An individual takes 300 mg of the drug initially. After 2 hours, about 223 mg remain in the bloodstream. Approximately how many mg of the drug remain in the individual's bloodstream after 6 hours? 146 mg O 123.2174 mg O 103.4288 mg O 69 mgThe volume of a substance, A, measured in cubic centimeters increases according to the exponential growth model _A = 0.34, where t is measured in hours. The volume of another substance, B, also measured in cubic centimeters increases at a constant dt rate of 1 cm per hour according to the linear model _D =1. Att = 0, substance A has a volume A(0) = 3 and substance B has size B(0) = 5. At what time will both substances have the same volume? dt O 0.8614 hours O 0.9776 hours O 3.0303 hours 3.4531 hoursNewton's Law of Cooling states that the rate of change of the temperature of an object, 7, is proportional to the difference of 7 and the temperature of the region, TR or UP =k(T -To). An object with core temperature of 1200'F is removed from a fire and dit placed in a region with a constant temperature of 80'F. After 1 hour, its core temperature is 830'F. What is the object's core temperature 4 hours after it is taken off the fire? 370.4010'F 305.2214'F O 225.2214'F 145.4010 FBacteria in a culture increase at a rate proportional to the amount present. At time t = 0, there were 650 bacteria present. At time t = 2, there were 1,950 bacteria present. Which of the following equations describes the exponential model in terms of t? O In(3) y = 650e 2 In(2), y = 650e 3 y = 1950ez' y = 1950es