Question

1. [10 pts.] Solve the following differential equation (dfrac{dx}{dt} = dfrac{t}{x^2}.) 2. [15 pts.] The population of a certain species of fish living in a large lake at time t can be modeled using Verhulst's equation, otherwise known as the logistic equation (dfrac{dP}{dt} = P(eta - sigma P)) where P(t) is the population of fish measured in tens of thousands, and eta and sigma are constants representing the birth and death rates of the fish living in the lake, respectively. Suppose that eta = 1, sigma = 10^-3, and the initial population is N = 800. Solve this initial value problem and interpret the results in physical terms.

          1. [10 pts.] Solve the following differential equation

(dfrac{dx}{dt} = dfrac{t}{x^2}.)

2. [15 pts.] The population of a certain species of fish living in a large lake at time t can be modeled using Verhulst's equation, otherwise known as the logistic equation

(dfrac{dP}{dt} = P(eta - sigma P))

where P(t) is the population of fish measured in tens of thousands, and eta and sigma are constants representing the birth and death rates of the fish living in the lake, respectively. Suppose that eta = 1, sigma = 10^-3, and the initial population is N = 800. Solve this initial value problem and interpret the results in physical terms.

$1. [10 pts.] Solve the following differential equation (dfracdxdt = dfractx^2.) 2. [15 pts.] The population of a certain species of fish living in a large lake at time t can be modeled using Verhulst's equation, otherwise known as the logistic equation (dfracdPdt = P(eta - sigma P)) where P(t) is the population of fish measured in tens of thousands, and eta and sigma are constants representing the birth and death rates of the fish living in the lake, respectively. Suppose that eta = 1, sigma = 10^-3, and the initial population is N = 800. Solve this initial value problem and interpret the results in physical terms.$