Problem 1
Verifying Solutions of Differential Equations Verify that the function \(y=e^{-3 x}+2 x+3\) is a solution to the differential equation \(y^{\prime}+3 y=6 x+11\)
Problem 1
Put each of the following first-order linear differential equations into standard form. Identify \(p(x)\) and \(q(x)\) for each equation. a. \(y^{\prime}=3 x-4 y\) b. \(\frac{3 x y^{\prime}}{4 y-3}=2(\) here \(x>0)\) c. \(y=3 y^{\prime}-4 x^{2}+5\)
Problem 1
Using Separation of Variables Find a general solution to the differential equation \(y^{\prime}=\left(x^{2}-4\right)(3 y+2)\) using the method of separation of variables.
Problem 1
Create a direction field for the differential equation \(y^{\prime}=x^{2}-y^{2}\) and sketch a solution curve passing through the point \((-1,2)\).
Problem 2
A population of rabbits in a meadow is observed to be 200 rabbits at time \(t=0\). After a month, the rabbit population is observed to have increased by \(4 \%\). Using an initial population of 200 and a growth rate of \(0.04\), with a carrying capacity of 750 rabbits, a. Write the logistic differential equation and initial condition for this model. b. Draw a slope field for this logistic differential equation, and sketc h the solution corresponding to an initial population of 200 rabbits. c. Solve the initial-value problem for \(P(t)\). d. Use the solution to predict the population after 1 year.
Problem 2
Put the equation \(\frac{(x+3) y^{\prime}}{2 x-3 y-4}=5\) into standard form and identify \(p(x)\) and \(q(x)\).
Problem 2
Use the method of separation of variables to find a general solution to the differential equation \(y^{\prime}=2 x y+3 y-4 x-6\)
Problem 2
Identifying the Order of a Differential Equation What is the order of each of the following differential equations? a. \(y^{\prime}-4 y=x^{2}-3 x+4\) b. \(x^{2} y^{\prime \prime \prime}-3 x y^{\prime \prime}+x y^{\prime}-3 y=\sin x\) c. \(\frac{4}{x} y^{(4)}-\frac{6}{x^{2}} y^{\prime \prime}+\frac{12}{x^{4}} y=x^{3}-3 x^{2}+4 x-12\)
Problem 3
For the following problems, consider the logistic equation in the form \(P^{\prime}=C P-P^{2} .\) Draw the directional field and find the stability of the equilibria.\(C=3\)
Problem 3
Consider the differential equation \(y^{\prime}=f(x, y) .\) An equilibrium solution is any solution to the differential equation of the form \(y=c\), where \(c\) is a constant.
