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Exercise 8

This document discusses first-order linear differential equations. It defines these equations as having the form a(x)y' + b(x)y = c(x), where a(x), b(x), and c(x) are arbitrary functions of the independent variable x. The document provides examples of first-order linear and nonlinear differential equations. It also explains how to put a first-order linear differential equation into standard form by dividing both sides by the coefficient of y', resulting in the equation y' + p(x)y = q(x). This standard form will be used to find the general solution to such equations.

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78 views2 pages

Exercise 8

This document discusses first-order linear differential equations. It defines these equations as having the form a(x)y' + b(x)y = c(x), where a(x), b(x), and c(x) are arbitrary functions of the independent variable x. The document provides examples of first-order linear and nonlinear differential equations. It also explains how to put a first-order linear differential equation into standard form by dividing both sides by the coefficient of y', resulting in the equation y' + p(x)y = q(x). This standard form will be used to find the general solution to such equations.

Uploaded by

Malcolm
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Download as DOCX, PDF, TXT or read online on Scribd
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he differential equation in this initial-value problem is an example of a first-order linear differential

equation. (Recall that


a differential equation is first-order if the highest-order derivative that appears in the equation is 1.) In
this section, we
study first-order linear equations and examine a method for finding a general solution to these types of
equations, as well as
solving initial-value problems involving them.
Definition
A first-order differential equation is linear if it can be written in the form
a(x)y′ + b(x)y = c(x), (4.14)
where a(x), b(x), and c(x) are arbitrary functions of x.
Remember that the unknown function y depends on the variable x; that is, x is the independent variable
and y is the
dependent variable. Some examples of first-order linear differential equations are
⎛⎝
3x2 - 4⎞
⎠y′ + (x - 3)y = sinx
(sinx)y′ - (cosx)y = cot x
4xy′ + (3lnx)y = x3 - 4x.
Examples of first-order nonlinear differential equations include
⎛⎝
y′⎞ ⎠4 - ⎛ ⎝y′⎞ ⎠3 = (3x - 2)⎛ ⎝y + 4⎞ ⎠
4y′ + 3y3 = 4x - 5
⎛⎝
y′⎞ ⎠2 = siny + cosx.
These equations are nonlinear because of terms like ⎛ ⎝y′⎞ ⎠4, y3, etc. Due to these terms, it is
impossible to put these
equations into the same form as Equation 4.14.
Standard Form
Consider the differential equation
⎛⎝
3x2 - 4⎞
⎠y′ + (x - 3)y = sinx.
Our main goal in this section is to derive a solution method for equations of this form. It is useful to have
the coefficient of
y′ be equal to 1. To make this happen, we divide both sides by 3x2 - 4.
y′ + ⎛
⎝
x-3
3x2 - 4
⎞⎠
y = sinx
3x2 - 4
This is called the standard form of the differential equation. We will use it later when finding the
solution to a general
first-order linear differential equation. Returning to Equation 4.14, we can divide both sides of the
equation by a(x). This
leads to the equation
(4.15)
y′ + b(x)
a(x)y =
c(x)
a(x).
Now define p(x) = b(x)
a(x) and q(x) = ac((xx)). Then Equation 4.14 becomes
y′ + p(x)y = q(x). (4.16)
We can write any first-order linear differential equation in this form, and this is referred to as the
standard form for a firstorder linear differential eq

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