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Exam 9: Differential Equations

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Question 31

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Use Euler's method with given values of n to obtain an approximation of the initial value problem when . Round your answers to four decimal places, if necessary. ​ , ​ , __________ ​ , __________

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Question 32

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Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The function is a solution of the differential equation .

A) True
B) False

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Question 33

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According to Lambert's law of absorption, the percentage of incident light L, absorbed in passing through a thin layer of material x, is proportional to the thickness of the material. If in. of a certain material reduces the light to half of its intensity, how much additional material is needed to reduce the intensity to one-fourth of its initial value?

Newton's law of cooling states that the rate at which the temperature of an object changes is directly proportional to the difference in temperature between the object and that of the surrounding medium. A cup of coffee is prepared with boiling water (212 °F) and left to cool on the counter in a room where the temperature is 70 °F. If the temperature of the coffee is 120 °F after 1 min, determine when the coffee will be cool enough to drink (say, 111 °F). Round the answer to the nearest tenth, if necessary. t = __________ min

Find the general solution of the first-order differential equation by separating variables. ​

Find the solution of the initial value problem. ​

Find the solution of the initial value problem. ​

y is a solution of the differential equation.n ​ any constant; ​

Assume that the rate of change of the unit price of a commodity is proportional to the difference between the demand and the supply so that ​ where k is a constant of proportionality. Suppose that D=74-5p, S=2+3p, and p(0)=2. Find a formula for p(t).

The rate at which the concentration of a drug in the bloodstream decreases is proportional to the concentration at any time t. Initially, the concentration of the drug in the bloodstream is B₀ g/mL. What is the concentration of the drug in the bloodstream at any time t? Formulate but do not solve the problem in terms of a differential equation with a side condition.

Find the general solution of the first-order differential equation by separating variables. ​

Use Euler's method with given values of n to obtain an approximation of the initial value problem when . Round your answers to four decimal places, if necessary. ​ , ​ , __________ ​ , __________

A tank initially contains 60 gal of brine, in which 10 lb of salt is dissolved. Brine containing 2 lb of dissolved salt per gallon flows into the tank at the rate of 4 gal/min, and the well-stirred mixture flows out of the tank at the same rate. How much salt is present in the tank at the end of 10 min?Round the answer to the nearest tenth, if necessary.__________ lb

An amount of money deposited in a savings account grows at a rate proportional to the amount present. (It can be shown that an amount of money grows in this manner if it earns interest compounded continuously.) Suppose $50,000 is deposited in a fixed account earning interest at the rate of 8%/year compounded continuously. What is the accumulated amount after 6 yr? Round the answer to the nearest cent. $__________ How long does it take for the original deposit to double in value? Round the answer to the nearest tenth, if necessary. __________ yr

Find the general solution of the first-order differential equation by separating variables. ​

Determine whether the statement is true or false. If it is true, explain why it is true. If it is false, give an example to show why it is false. The function is a solution of the differential equation .

Find the solution of the initial value problem. ​