Mastering Initial Value Problems in Differential Equations

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Which of the following best describes an ordinary differential equation?

An equation involving known constants and their derivatives

An equation involving known functions and their derivatives

An equation involving an unknown constant and its derivatives

An equation involving an unknown function and its derivatives (correct)

What is the order of the differential equation dy/dt = y - t?

Fourth order

Second order

Third order

First order (correct)

Which of the following equations is a second-order differential equation?

$y' = y^2 - ty$

$yy'' + t^2y = \cos(t)$ (correct)

$y = y^2 - t$

$yy' + 4y = e^{-3t}$

What will be covered in this chapter according to Section 2.1?

Exact solutions and their applications (D) Signup and view all the answers

What is the unknown function in the equation dy/dt = y - t?

y (C) Signup and view all the answers

Which of the following equations is a first-order differential equation?

$y' = y^2 - ty$ (B) Signup and view all the answers

What is the order of the differential equation $\frac{{\partial^2w}},{{\partial t^2}} = c^2\frac{{\partial^2w}},{{\partial x^2}}$?

Second order (C) Signup and view all the answers

Which of the following is the definition of an initial value problem?

A differential equation with an initial condition (C) Signup and view all the answers

What is the interval of existence of the solution to the initial value problem $y' = y^2$, $y(0) = 1$?

$(-\infty, \infty)$ (A) Signup and view all the answers

What is the general solution of the differential equation $y' = x + y$?

$y(x) = -1 - x + Ce^{-x}$ (D) Signup and view all the answers

What is the interval of existence of the solution to the differential equation $s' = \sqrt{r}$?

$[0, \infty)$ (A) Signup and view all the answers

Verify that $x(s) = 2 - Ce^{-s}$ is a solution of the differential equation $x' = 2 - x$. What is the solution that satisfies the initial condition $x(0) = 1$?

$x(s) = 2 - e^{-s}$ (C) Signup and view all the answers

What is the geometric interpretation of a differential equation?

The slope of the solution curve at a given point (B) Signup and view all the answers

What is a direction field in the context of a differential equation?

A small line segment attached to each point in the solution curve (A) Signup and view all the answers

Which type of differential equation involves partial derivatives of an unknown function of more than one independent variable?

Partial differential equation (D) Signup and view all the answers

What is the normal form for a first-order differential equation?

$y(t) = f(t, y)$ (D) Signup and view all the answers

How can we determine if a given function is a solution to a differential equation?

Substitute the function and its derivative(s) into the equation and check for equality (D) Signup and view all the answers

What is the general solution to the first-order equation $y'(t) = -2ty$?

$y(t) = Ce^{-t^2}$ (C) Signup and view all the answers

What is the general solution to the equation $y^{(n)} = f(t, y, y', ..., y^{(n-1)})$?

$y(t) = Ce^{f(t, y, y', ..., y^{(n-1)})}$ (D) Signup and view all the answers

Is the function $y(t) = \cos(t)$ a solution to the differential equation $y' = 1 + y^2$?

No (A) Signup and view all the answers

If $y(t) = -\frac{1},{t - C}$ is a general solution of $y' = y^2$, what is the particular solution satisfying $y(0) = 1$?

$y(t) = -\frac{1},{t + 1}$ (A) Signup and view all the answers

Which of the following best describes the use of computer-generated direction fields in understanding differential equations?

They give a new interpretation of a solution (A) Signup and view all the answers

In the context of differential equations, what does the slope of the solution curve represent?

The value of f(t, y(t)) (B) Signup and view all the answers

What does finding a solution to a differential equation represent geometrically?

Finding a curve in the ty-plane that is tangent to the direction field at every point (B) Signup and view all the answers

Which equation represents the direction field shown in Figure 4?

$y' = y$ (C) Signup and view all the answers

What does the solution curve of $y' = y, y(0) = 1$ look like in relation to the direction field?

It is tangent to the direction field at every point (A) Signup and view all the answers

What is the slope of the solution curve of $y' = y, y(0) = 1$ at the point (0, 1)?

0 (B) Signup and view all the answers

What is the value of y(1) for the solution curve of $y' = y, y(0) = 1$?

2 (B) Signup and view all the answers

What is the value of y(2) for the solution curve of $y' = y, y(0) = 1$?

e (B) Signup and view all the answers

What is the general solution to the differential equation $y' = y$?

$y = e^t$ (B) Signup and view all the answers