Exam 10: Introduction to Differential Equations

Solve the initial value problem.

Consider the initial value problem A) Use Euler's method with to approximate . Give your answer to five decimal places. B) Solve the initial value problem and compute to five decimal places. C) Compute the error in the approximation of .

A rat population in a certain field is initially 200. After 4 years, the population increases to 350. Assuming logistic growth with a carrying capacity of 550, what is the rat population 3 years after the population reached 350?

A certain chemical dissolves in water at a rate proportional to the product of the amount of chemical that had not yet been dissolved and the difference between its concentration in a saturated solution and its current concentration. It is known that 100 g of the chemical are dissolved in 200 g of saturated solution. If 60 g of the chemical are added to 200 g of water, 20 g are dissolved in 2 h. How many grams of the chemical are dissolved in 4 h?

Use the following steps. A) Write an equation for the areas using integrals. B) Differentiate the equation in A and solve the resulting linear equation. Ans: A) B) -A pool contains 12 million gal of fresh water. Contaminated water flows into the pool at a rate of 4 million gal/year as the mixture in the pool flows out at the same rate. The concentration of the chemical in the incoming contaminated water is g/gal, where t is in years. Find the amount of the chemical in the pool at time .

Solving the differential equation , we obtain the solution:

Find all curves such that the tangent line at any point on the curve has a y-intercept that is equal to .

The horizontal cross sections at height of a tank are discs of radius . The height of the tank is 10 m. The tank is filled with water, and the water drains through a square hole with a side of 10 cm at the bottom of the tank. How long does it take for the tank to empty?

A deer population for a certain area is initially 500. After 3 years, the population increases to 600. Assuming logistic growth with a carrying capacity of 850, what is the deer population 2 years after the population reached 600?

Match the differential equation with its slope field. A) B) C) i) ii) iii)

Consider the logistic equation . The solution with the following initial condition is decreasing to minus infinity at a finite value of :

Match the differential equation with its slope field (A) (B) (C) i) ii) iii)

Solve the initial value problem .