周次: 9 日期: April 20, 2022 节次: 1
In this lecture, we introduce some basic concept for differential equations, how to think of them, how to describe them and how to solve them. The most important thing is to know how to solve a differential equation. We will solve typical separable differential equations as well as the first-order linear differential equations.
Chapter 9. Further Applications of Integration
Overview.
Previously we’ve learned anti-derivative. For a continuous function
Another view for the concept of anti-derivative, is that we are solving an equation related with function
Any equation involving the derivative for unknown functions can be called a differential equation. Therefore what we did, was to solve a simplest differential equation.
In this chapter, we study several more involved differential equations, having the form
where
§ 9.1 Slope Fields and Separable Differential Equations
General First-Order Differential Equations and Solutions.
Definition. A First-order differential equation is an equation
in which
The term “first-order”, refers to the highest order of derivative is having order
Definition. A Solution to equation (1) is a differentiable function
on that interval.
Definition. A general solution to equation (1) is a solution that contains all possible solutions. It may contain arbitrary constants, so that when constants are being fixed, then we get a particular solution.
Example 1. Verifying Solution Functions
Show that every member of the family of functions
is a solution of the first-order differential equation
Sometimes we need a particular solution rather than the general one to a differential equation
The first-order initial value problem is a differential equation
Example 2. Verifying That a Function Is a Particular Solution
Show that the function
is a solution to the first-order initial value problem
Slope Fields: Viewing Solution Curves
Any particular solution to a differential equation can be viewed as a graph, a plane curve, at every point
Indeed, before solving the differential equation, we could illustrate the slope for each point on the plane by line segments. Each line segment has the slope indicating by
Slope Field and General Solutions
Here are some of the slope fields for particular
For Example 1
For Example 2
Separable Equations
In the equation
For our convenience, let’s rewrite it in the form
Moving terms with different variables to different sides, then we get
Then we simply integrate both sides of this equation:
Equation (2) indication an implicit relation between
The Justification
To justify the procedure, we use the differential relation
Example 3. Solving a Separable Equation
Solving the differential equation
Example 4. Solve the equation
Example 5. The Initial Value Problem
Torricelli’s Law
If you drain a tank, the rate at which the water runs out is a constant times the square root of the water’s depth
Example 6. Drain a Tank
A right circular cylindrical tank with radius
§ 9.2 First-Order Linear Differential Equations
Differential equations are, in general, difficult or even impossible to solve explicitly. However, for some particular classes, for example: linear differential equations, we can show that it is solvable.
In this section, we are looking at a simplest type of linear differential equations, which can be solved explicitly. The type we are looking at is called First-order Linear Differential Equations.
Definition (First-Order Linear Differential Equations)
A differential equation
where
Remark 1. ****
- What does first-order mean?
First-order refers to the highest order derivative appeared in the equation is first-derivative.
- What does linear mean?
Linear means all the operations occurring on
is a linear operator. Which means, for any linear combination
For example,
Remark 2.
A linear DE is much easier to solve. Suppose
then
Example 1. Finding the Standard Form
Put the following equation in standard form:
Solving Linear Equations
We solve the equation
by transforming the left side into the derivative of certain product function
Why does multiplying by v(x) work?
Suppose by multiplying a function
Solution of equation (3) was expressed in terms of the function
How such v(x) is found?
If step (4) can be transformed into step (5) successfully, then
Therefore
It is another first-order linear DE which can be solved by assuming
Summary
Example 2. Solving a First-Order Linear Differential Equation
Solve the equation
Solution. (
Example 3. Solving a First-Order Linear Initial Value Problem
Solve the equation
given the initial condition
Solution. (
Example 4. Find the particular solution of
satisfying
Solution. (
Note.
If the function
the linear equation is separable:
Then the DE can be solved by integration.
Applications.
RL Circuits
An electrical circuit whose resistance is
Ohm’s law for such a circuit has to be modified. The modified form is
where
Example 5. Electric Current Flow
The switch in the RL circuit is closed at time