Modeling with first order Differential Equations (Five Problems).

9. When a vertical beam of light passes through a transparent medium, the rate at which its intensity I decreases is proportional to I(t). where t represents the thickness of the medium (in feet). In clear seawater, the intensity 3 feet below the surface is 25% of the initial intensity Jo of the incident beam. What is the intensity of the beam 15 feet below the surface?

19. A tank contains 200 liters of fluid in which 30 grams of salt is dissolved. Brine containing I gram of salt per liter is then pumped into the tank at a rate of 4 L/min; the well-mixed solution is pumped out at the same rate. Find the number A(t) of grams of salt in the tank at time t.

20. Solve Problem 19 assuming that pure water is pumped into the tank.

29. A 100-volt electromotive force is applied to an RC series circuit in which the resistance is 200 ohms and the capacitance is 10^-4 farad. Find the charge q(t) on the capacitor if q(0)=0. Find the current i(t).

30. A 200-volt electromotive force is applied to an RC series circuit in which the resistance is 1000 ohms and the capacitance is 5 X 10^-6 farad. Find the charge q(t) on the capacitor if q(0)=0. Find the current i(t).

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First-order differential equations are used to model word problems. The solution is detailed and well presented. The response received a rating of "5/5" from the student who originally posted the question.

Please show the steps in the attached problems so that I can better understand what I am learning.
Q1 is a Bernoulli equation, primes denote derivatives with respect to x.
Please only complete questions 1 and 2.
1. Find the general solution to 2xy' + y^3 * exp(-2x) = 2xy.
2. Find the general solution of reducible sec

Please find the attached.
i) x ̇=-(t+1)x/t
ii) x ̇=-t^3/(x+1)^2
a) Find all the solutions of i) and ii).
b) for i) and ii) Discuss/Give possible choices for a domain B⊂R^2 such that the IVP with x(t_0 )=x_0 has a unique solution.

For each of the following ordinairy differentialequations, indicate its order, whether it is linear or nonlinear, and whether it is autonomous or non-autonomous.
a) df/dx +f^2=0
(See attachment for all questions)

2. The solution of the initial value problem (see attachment) is known to exist on -infinite < t differential equation y' + theta*y = 1, where theta is a real number. For what values of theta is lim y(t) finite? What is the limit in this case

Differential Equation (IX): Formation of DifferentialEquations by Elimination
Eliminate the arbitrary constants from the equation: y = Ae^x + Be^2x + Ce^3x. Make sure to show all of the steps which are involved.

Two springs are attached in series as shown in Figure 5.42. If the mass of each spring is ignored, show that the effective spring constant k ot the system is defined by I/k = I/k + I/k2.
A mass weighing W pounds stretches a spring 1/2 foot and stretches a different spring 1/4 foot. The two springs are attached, and the mass is

Hi,
Please help working on
section 1.1 problems 2,4,8,14,16
section 1.2 problems 6,10,20,24,27
thank you
See attached
Classify each as an ordinary differential equation (ODE) or a partial differential equation (PDE), give the order, and indicate the independent and dependent variables. If the equation is an ODE, ind

Differential Equation (XII)
Formation of DifferentialEquations by Elimination
Find the differentialequations of all parabolas whose axes are parallel to the axis of y.

Please show all work with explanations.
Thank you for your help.
4. Find
6. Find the arc length of the graph of the function over the interval .
18. Solve the differential equation