13. Intro to Differential Equations

33–42. Solving initial value problems Solve the following initial value problems.

y'(t) = 1 + eᵗ, y(0) = 4

9–14. Growth rate functions Make a sketch of the population function P (as a function of time) that results from the following growth rate functions. Assume the population at time t = 0 begins at some positive value.

Consider the differential equation $\frac{dy}{dx}=xy+x$. Which of the following best describes this equation?

What is the general solution to the differential equation $\frac{dy}{dx}=x-13y^2$ for $y>0$?

Solve the differential equation $y^{\text{'}\text{'}}+y=\sin \left(x\right)$ using the method of variation of parameters. Which of the following is the general solution?

Find the general solution of the second-order differential equation $3y^{″}+2y^{\prime }+y=0$.

Solve the initial-value problem for the homogeneous differential equation: $(x^2+2y^2)\frac{dx}{dy}=xy$, with $y(-1)=3$. What is the explicit solution for $x$ in terms of $y$?

Solve the differential equation $\frac{dy}{dx}=2y$ by separation of variables. Which of the following is the general solution?

Solve the differential equation $y^{\text{'}\text{'}}+2y^{\text{'}}=2x+7-e^{-2x}$ using the method of undetermined coefficients. Which of the following is the general solution?

Solve the differential equation $y^{\mathrm{\text{'}\text{'}}}+2y^{\mathrm{\text{'}}}=2x+3-e^{-2x}$ using the method of undetermined coefficients. Which of the following is a particular solution?

Logistic growth parameters A cell culture has a population of 20 when a nutrient solution is added at t=0. After 20 hours, the cell population is 80 and the carrying capacity of the culture is estimated to be 1600 cells.

c. After how many hours does the population reach half of the carrying capacity

Solve the differential equation $e^{x}y\frac{dy}{dx}=e^{-y}+e^{-3x}-y$ by separation of variables.

Solve the differential equation $x\frac{dy}{dx}=6y$ by separation of variables.

In Exercises 1–4, show that each function y=f(x) is a solution of the accompanying differential equation.

1. 2y' + 3y = e^(-x)

a. y = e^(-x)

Which of the following is the solution to the differential equation $\frac{dy}{dx}=4xy$, where $y\left(2\right)=-2$?