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Chapter 1: Problem 2

What is the order of the differential equation? $$ t^{4} y^{\prime}+y \sin t=6 $$

Short Answer

Expert verified
Answer: The order of the given differential equation is 1.

Step by step solution

01

Identify the highest order of the derivative

In the given differential equation, \(t^4 y' + y \sin t = 6\). Identify the highest order of the derivative present in the equation. We have only one derivative term, \(t^4 y'\).
02

Determine the order of the highest order derivative

We see that \(t^4 y'\) is the first derivative of the function \(y\) with respect to \(t\). Therefore, the highest order derivative is of order 1.
03

Write the final result

Since the highest order derivative present in the differential equation is of order 1, the order of the differential equation is 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Order of Differential Equation
The order of a differential equation is determined by identifying the highest order of the derivative present in the equation. In mathematical terms, this means locating where the derivative with the largest "exponentiation" or "differentiation" occurs in the entire equation. For example, if you have an equation like \( t^4 y' + y \sin t = 6 \), you will look for the highest order of the derivative of \( y \).
  • The order is crucial because it indicates the number of initial or boundary conditions required to find a unique solution to the equation.
  • Higher order equations may describe more complex physical systems or phenomena.

In the provided example, the term \( t^4 y' \) represents the first derivative of \( y \) with respect to \( t \). Since this is the only derivative term in the equation, the order of this particular differential equation is 1.
First Order Differential Equation
A first order differential equation involves only the first derivative of the function and no higher derivatives. They appear frequently in real-world applications such as population dynamics, cooling/heating laws, or simple electrical circuits. Let's break down what makes an equation first order:
  • It contains terms like \( y' \) (the first derivative of \( y \)), but not \( y'' \) or any higher derivatives.
  • The order provides insight into the number of conditions needed; for first order equations, you typically need one initial condition.

The equation \( t^4 y' + y \sin t = 6 \) is a perfect example of a first order differential equation as the highest derivative present is the first derivative \( y' \). This means it could model phenomena that can be described by just knowing an initial value, like the initial temperature distribution of an object.
Derivatives in Differential Equations
Derivatives are a key component of differential equations. They represent rates of change or how a system evolves over time. Let's explore what derivatives tell us in differential equations:
  • First derivatives (like \( y' \)) indicate how fast something changes; for instance, speed is the first derivative of position with respect to time.
  • Higher order derivatives (\( y'', y''', \) etc.) can represent changes in rates, such as acceleration being the second derivative of position.

In a differential equation, derivatives help describe dynamic systems and allow us to model and predict their behavior under various conditions. In the equation \( t^4 y' + y \sin t = 6 \), the derivative \( y' \) tells us about the rate of change of \( y \) with respect to \( t \), highlighting how quantities are interconnected and evolved in such systems.

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Most popular questions from this chapter

(a) State whether or not the equation is autonomous. (b) Identify all equilibrium solutions (if any). (c) Sketch the direction field for the differential equation in the rectangular portion of the \(t y\)-plane defined by \(-2 \leq t \leq 2,-2 \leq y \leq 2\). $$ y^{\prime}=\sin y $$

For what value(s) of the constant \(k\), if any, is \(y(t)\) a solution of the given differential equation? $$ y^{\prime}+(\sin 2 t) y=0, \quad y(t)=e^{k \cos 2 t} $$

What is the order of the differential equation? $$ \left(y^{\prime \prime \prime}\right)^{4}-\frac{t^{2}}{\left(y^{\prime}\right)^{4}+4}=0 $$

Use the result of Exercise 16 to solve the initial value problem. $$ y^{\prime \prime}-4 y=0, \quad y(0)=2, \quad y^{\prime}(0)=0 $$

(a) State whether or not the equation is autonomous. (b) Identify all equilibrium solutions (if any). (c) Sketch the direction field for the differential equation in the rectangular portion of the \(t y\)-plane defined by \(-2 \leq t \leq 2,-2 \leq y \leq 2\). $$ y^{\prime}=-y+1 $$
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