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Chapter 20: Problem 1

Give one example each of a first-order linear equation, first-order non-linear equation, second-order linear equation, second-order non-linear equation.

Short Answer

Expert verified
Examples: \( 2x + 3 = 0 \), \( \ln(x) + 2 = 0 \), \( 3x^2 + 2x - 1 = 0 \), \( x^2 + \sin(x) = 0 \).

Step by step solution

01

Identifying First-Order Linear Equation

A first-order linear equation can be expressed in the form \( ax + b = 0 \). For instance, the equation \( 2x + 3 = 0 \) is a first-order linear equation because the highest power of \( x \) is 1, and it follows the general form.
02

Identifying First-Order Non-Linear Equation

A first-order non-linear equation doesn't strictly follow the form of \( ax + b = 0 \). An example is \( \ln(x) + 2 = 0 \), where the term \( \ln(x) \) makes the equation non-linear.
03

Identifying Second-Order Linear Equation

A second-order linear equation typically takes the form \( ax^2 + bx + c = 0 \). For example, \( 3x^2 + 2x - 1 = 0 \) is a second-order linear equation as the highest degree of \( x \) is 2.
04

Identifying Second-Order Non-Linear Equation

A second-order non-linear equation involves terms that make the equation non-linear despite the second-degree variable. An example is \( x^2 + \sin(x) = 0 \), where the \( \sin(x) \) makes it non-linear.

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Key Concepts

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

First-Order Linear Equation
A first-order linear equation is one of the building blocks in understanding differential equations. This type of equation involves a single variable and can be written in the standard form \( ax + b = 0 \), where \( a \) and \( b \) are constants. The defining characteristic is the highest power of the variable is one, making it linear.

Consider the equation \( 2x + 3 = 0 \). It is linear because:
  • The variable \( x \) is to the first power.
  • There are no products or powers greater than one involving \( x \).
Solving it involves simple arithmetic: subtract 3 from both sides and divide by 2, thus providing a straightforward solution. This simplicity makes first-order linear equations an essential tool for modeling real-world linear relationships.
First-Order Non-Linear Equation
A first-order non-linear equation is characterized by not fitting the linear equation format. Any term that includes a logarithm, exponential, or a variable raised to a power different from one can render an equation non-linear.

For example, take \( \ln(x) + 2 = 0 \). This equation involves the natural logarithm of \( x \), which is a non-linear operation:
  • The presence of \( \ln(x) \) disrupts the linearity.
  • Non-linear equations often correspond to real-world scenarios with more complex relationships.
Solving first-order non-linear equations can be more complex than their linear counterparts, often requiring specific algebraic techniques or numerical methods for a solution.
Second-Order Linear Equation
Second-order linear equations extend the linear concept into higher dimensions, where the highest power of the variable is two. Typically expressed in the form \( ax^2 + bx + c = 0 \), these equations can describe numerous phenomena from physics to engineering.

For example, consider \( 3x^2 + 2x - 1 = 0 \). Key features include:
  • The variable's highest power is two, leading to a quadratic form.
  • The equation remains linear since the powers of each term respect the criteria for linearity.
Solutions often involve techniques like factoring, completing the square, or using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). These elements provide the groundwork for more extensive applications in scientific computations.
Second-Order Non-Linear Equation
Second-order non-linear equations introduce a layer of complexity that arises from non-linear terms in conjunction with quadratic expressions. They are represented by equations like \( x^2 + \sin(x) = 0 \), combining polynomial and non-linear functions.

Take this equation as a case in point:
  • The quadratic term \( x^2 \) appears alongside a non-linear sine function \( \sin(x) \).
  • The presence of trigonometrics or other non-linear terms breaks the linear structure.
Solving these equations demands advanced methods, potentially including graphical solutions or iterative numerical techniques, to capture the complex behaviors they can model.

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Most popular questions from this chapter

Integrate the equation \(\frac{\mathrm{d} y}{\mathrm{~d} x}=3 x^{2}\) subject to the condition \(y(1)=4\) in order to find the particular solution.

Two tanks containing a liquid are placed in series so that the first discharges into the second and the second discharges into a waste outlet. Let \(q_{1}(t)\) and \(q_{2}(t)\) be the flow rates out of the two tanks respectively, and let the height of liquid in each of the tanks be \(h_{1}(t)\) and \(h_{2}(t)\) respectively. The two tanks are identical and each has a constant cross- sectional area \(A\). The outflow from each tank is proportional to the height of liquid in the tank. At \(t=0\) the height of liquid in the first tank is \(h_{0}\) and the second tank is empty. (a) Derive and solve the differential equation for \(h_{1}(t)\). (b) Hence find \(q_{1}(t)\). (c) Derive and solve the differential equation for \(h_{2}(t)\). (d) Hence find \(q_{2}(t)\).

Find the general solution of the equation $$ \frac{\mathrm{d} y}{\mathrm{~d} t}+(\tan t) y=\cos t $$

Find the general solution of the equation $$ \frac{\mathrm{d} x}{\mathrm{~d} t}=t(x-2) $$ Find the particular solution that satisfies the condition \(x(0)=5\)

Find a first-order equation satisfied by \(x=A \mathrm{e}^{-2 t}\).
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