Simple Differential Equation (Ordinary Differential Equation)

In summary, the order of the ODE is 1st order and to check if a function solves the equation, you do not need to actually solve the equation, but rather substitute the function into the equation and see if it satisfies it. To solve the equation, you can use an integrating factor and then solve for the function. Alternatively, you can use Bernoulli's equation and consider the constant 5e^{-2x} as C.
  • #1
eskie
5
0
Problem:
y'+2y=4(x+1)2 ----> y=5e-2x+2x2+2x+1

1. What the Order of the ODE?
It's 1st order

2 How do you check whether a particular function solves an equation?
If you solve y'+2y=4(x+1)2 and make it y=5e-2x+2x2+2x+1. I want the whole solution... thanks...
 
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  • #2
Redo the whole thing:

1) What is the order of the ODE?

2) How do you check whether a particular function solves an equation?
 
  • #3
eskie said:
Problem:
y'+2y=4(x+1)2 ----> y=5e-2x+2x2+2x+1

1. What the Order of the ODE?
It's 1st order
Yes, that is correct.

2 How do you check whether a particular function solves an equation?
If you solve y'+2y=4(x+1)2 and make it y=5e-2x+2x2+2x+1. I want the whole solution... thanks...
How would you determine whether x= 7 satisifies the equation [itex]x^9+ 9x^3- 2x^2+ 3x- 4= 0[/itex]? Not by actually solving the equation! Just put x= 7 into the equation and see if it makes the equation true or not.

Same thing here. You would NOT need to actually solve the equation. Just calculate the derivative of y: [itex]y' = -10e^{-2x}+ 4x+ 2[/itex] and put it and [itex]y= 5e^{-2x}+ 2x^2+ 2x+ 1[/itex] into the equation: [itex]y'+ 2y= -10e^{-2x}+ 4x+ 2+ 10e^{-2x}+ 4x^2+ 4x+ 2[/itex] is that equal to 4(x+ 1)?
 
  • #4
oh i see... how about integrate the y' of y'+2y=4(x+1)2?
 
  • #5
eskie said:
oh i see... how about integrate the y' of y'+2y=4(x+1)2?
That would be the way to find the solution, rather than verifying that a given function is the solution!

To find a solution is generally a lot harder to do than verifying that a function is a solution (or not).
 
  • #6
oh... i see... i just to solve that equation just like this...
y'=4x
dy/dx=4x
dy=4xdx
y=2x2
 
  • #7
eskie said:
oh... i see... i just to solve that equation just like this...
y'=4x
dy/dx=4x
dy=4xdx
Not quite!

In your original ODE, that approach won't help you much.

You need to use what we call an "integrating factor" here!

Let g(x)=e^{2x}y(x}

Then, we have:
[tex]\frac{dg}{dx}=e^{2x}(y'+2y)[/tex]

Note that the expression in the parenthesis is the left-hand side of your ODE, so that you may write:
[tex]\frac{dg}{dx}=4e^{2x}(x+1)^{2}[/tex]

THIS ODE is separable, and you can proceed to solve for g(x) first, nd then for y(x)..
 
  • #8
oh i see... thanks anyway... :) our teacher said that we must use bernoulli's eq. to solve the eq... and the 5e-2x is constant.. therefore i can just change it to C
 

1. What is a simple differential equation (ordinary differential equation)?

A simple differential equation, or ordinary differential equation (ODE), is a mathematical equation that describes the relationship between a function and its derivatives. It involves a single independent variable and one or more dependent variables. ODEs are commonly used in science and engineering to model dynamic systems and predict their behavior over time.

2. What is the difference between a simple differential equation and a partial differential equation?

The main difference between a simple differential equation and a partial differential equation (PDE) is the number of independent variables. A simple differential equation has only one independent variable, while a PDE has two or more independent variables. This means that a PDE involves partial derivatives, while an ODE only involves ordinary derivatives. PDEs are typically used to model systems that vary in space as well as time, while ODEs are used for systems that only vary with time.

3. What are the different types of simple differential equations?

There are several types of simple differential equations, including linear and non-linear, first-order and higher-order, and homogeneous and non-homogeneous. A linear ODE is one in which the dependent variable and its derivatives appear in a linear form, while a non-linear ODE is one in which they appear in a non-linear form. First-order ODEs involve only the first derivative of the dependent variable, while higher-order ODEs involve higher derivatives. Homogeneous ODEs have a dependent variable and its derivatives on one side of the equation, while non-homogeneous ODEs have additional terms on the other side.

4. What are some real-world applications of simple differential equations?

ODEs have many real-world applications, including in physics, chemistry, biology, economics, and engineering. For example, they can be used to model the growth of a population, the motion of a pendulum, the diffusion of particles in a liquid, and the spread of a disease. In engineering, ODEs are used to design control systems for airplanes, cars, and other machines. They are also used in financial modeling to predict stock prices and interest rates.

5. How are simple differential equations solved?

The solution to a simple differential equation is a function that satisfies the equation and any given initial conditions. Depending on the type of ODE, different methods can be used to find the solution. These include analytical methods such as separation of variables, substitution, and variation of parameters, as well as numerical methods such as Euler's method, the Runge-Kutta method, and the finite difference method. The appropriate method to use depends on the complexity of the ODE and the accuracy required for the solution.

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