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Chapter 4: Problem 1

Determine the order of the following differential equations. $$ y^{\prime}+y=3 y^{2} $$

Short Answer

Expert verified
The order of the differential equation is 1.

Step by step solution

01

Identify the Differential Equation Components

The given differential equation is \( y^{\prime} + y = 3y^{2} \). Here, \( y^{\prime} \) is the first derivative of \( y \) with respect to an independent variable (typically \( x \)). The term \( y \) and on the right-hand side \( 3y^{2} \) involve no derivatives.
02

Determine the Highest Derivative

In the differential equation \( y^{\prime} + y = 3y^{2} \), identify the highest derivative present. The highest derivative in this equation is \( y^{\prime} \), which is the first derivative.
03

Establish the Order of the Differential Equation

The order of a differential equation is determined by the highest order of derivative present. Since the highest derivative in our equation \( y^{\prime} + y = 3y^{2} \) is the first derivative \( y^{\prime} \), the order of this 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 Equations
The order of a differential equation is a fundamental concept in understanding how derivatives play a role in mathematical models. It is defined by the highest derivative present in the equation.
For example, if an equation contains only the first derivative, such as in our equation \( y^{\prime} + y = 3y^{2} \), its order is 1. This means we are working with first-order processes.
  • **Identify the Highest Derivative:** Look for the term in the equation with the most prime symbols (\('\)). The more primes, the higher the derivative.
  • **Example:** In \( y^{\prime}+y=3y^{2} \), the highest derivative is \( y^{\prime} \), hence the equation is first order.
This is crucial in determining the methods used for solving the equation and understanding the behavior of the system it models.
First Derivative
A first derivative, denoted by \( y^{\prime} \), represents the instantaneous rate of change of the function \( y \) with respect to its independent variable, often \( x \).
In the differential equation \( y^{\prime} + y = 3y^{2} \), \( y^{\prime} \) is explicitly present, highlighting its role as the primary factor of change.
  • **Exploring Derivatives:** The derivative tells us how functions slope or curve as their inputs change.
  • **Instantaneous Change:** In simple terms, it measures how steeply \( y \) is rising or falling at any given moment.
Understanding this first derivative is key to grasping how different elements within an equation interact and how they influence the system modeled by the equation.
Mathematical Analysis
Mathematical analysis in the context of differential equations involves assessing and interpreting the solutions and their behaviors.
Here, it鈥檚 about understanding how alterations in \( y^{\prime}+y=3y^{2} \) influence solutions and their stability.
  • **Analyzing Solutions:** Determine how solutions behave over time, whether they grow, decay, or stabilize.
  • **System Behavior:** Investigate how slight changes in initial conditions affect the system's trajectory.
This analysis is not just about solving equations but understanding the underlying mechanics and implications of these solutions. It鈥檚 crucial for predicting future behavior in practical scenarios, from physics to engineering.

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Most popular questions from this chapter

[T] Rabbits in a park have an initial population of 10 and grow at a rate of \(4 \%\) per year. If the carrying capacity is 500 , at what time does the population reach 100 rabbits?

Draw the directional field associated with the differential equation, then solve the differential equation. Draw a sample solution on the directional field. $$ y^{\prime}=2 y-y^{2} $$

Assume an initial nutrient amount of \(I\) kilograms in a tank with \(L\). liters. Assume a concentration of \(c \mathrm{~kg} /\) \(\mathrm{L}\) being pumped in at a rate of \(r \mathrm{~L} / \mathrm{min}\). The tank is well mixed and is drained at a rate of \(r \mathrm{~L} / \mathrm{min}\). Find the equation describing the amount of nutrient in the tank.

Find the general solution to the differential equations. $$ y^{\prime}=x^{2}+3 e^{x}-2 x $$

Leaves accumulate on the forest floor at a rate of \(2 \mathrm{~g} / \mathrm{cm}^{2} / \mathrm{yr}\) and also decompose at a rate of \(90 \%\) per year. Write a differential equation goveming the number of grams of leaf litter per square centimeter of forest floor, assuming at time 0 there is no leaf litter on the ground. Does this amount approach a steady value? What is that value?
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