13. Intro to Differential Equations

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 $y\left(2\right)=-2$?

Explain how the growth rate function determines the solution of a population model.

7–16. Verifying general solutions Verify that the given function is a solution of the differential equation that follows it. Assume C, C1, C2 and C3 are arbitrary constants.

u(t) = C₁eᵗ + C₂teᵗ; u''(t) - 2u'(t) + u(t) = 0

Solve the following system of differential equations by systematic elimination:

$$\frac{\partial x}{\partial t}=2x-y$$$$\frac{\partial y}{\partial t}=x$$
Which of the following gives the general solution for $x\left(t\right)$?

State the order of the differential equation and indicate if it is linear or nonlinear. 

$\frac{d^{2}y}{d{x}^2}-\left(\frac{dy}{dx}\right)(1-x)=2$

The general solution of a first-order linear differential equation is y(t) = Ce⁻¹⁰ᵗ − 13. What solution satisfies the initial condition y(0) = 4?

Which of the following is the general solution to the differential equation $\frac{dy}{dx}=9x^2{y}^2$?

22–25. Equilibrium solutions Find the equilibrium solutions of the following equations and determine whether each solution is stable or unstable.

y′(t) = y(3+y)(y-5)

Integral Equations

In Exercises 7–12, write an equivalent first-order differential equation

and initial condition for y.

y = ln x + ∫ₓᵉ √ (t² + (y(t))²) dt

Write the formula for a logistic function that has values between y = 0 and y = 1, crosses the line y = 1/2 at x = 0, and has slope 5 at this point.

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.

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

y''(t) = teᵗ, y(0) = 0, y'(0) = 1

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

p'(x) = 2/(x² + x), p(1) = 0