Verifying a Solution for a Basic Differential Equation

In summary, the conversation is about verifying a solution for a given differential equation, specifically y = x-(x^-1). The person is struggling to separate the variables and asks for suggestions, but the other person clarifies that the task is only to verify the solution and not solve the equation.
  • #1
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Homework Statement
Show that y = x-(x^-1) is a solution for the differential equations:
xy' +y = 2x

The attempt at a solution
Frankly, I haven't a clue how to start.

If it weren't for the x in the xy' term this would be easy, since I could just integrate; unfortunately, I don't know how to separate the x's from the y's, and this is really a problem, because this is the first problem from the homework set. I re-read the textbook, but don't get it. Any suggestions for techniques to separate the variables?
 
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  • #2
The problem isn't asking you to solve the differential equation, merely to verify that the solution given is in fact a solution. So, substitute it in for y and y' and see if the equation holds true.
 

FAQ: Verifying a Solution for a Basic Differential Equation

1. What is a differential equation?

A differential equation is a mathematical equation that describes the relationship between a function and its derivatives. It represents how a quantity changes over time or space.

2. What is the difference between an ordinary and partial differential equation?

An ordinary differential equation involves only one independent variable, while a partial differential equation involves multiple independent variables. Ordinary differential equations are typically used to model physical systems with a single variable, while partial differential equations are used for systems with multiple variables.

3. How are differential equations used in science?

Differential equations are used in many areas of science, including physics, engineering, biology, and economics. They are used to model and analyze complex systems and predict how they will change over time. For example, they can be used to model population growth, chemical reactions, and the behavior of electric circuits.

4. What are the different types of differential equations?

The main types of differential equations are ordinary differential equations, partial differential equations, and stochastic differential equations. Ordinary differential equations can be further classified as linear or nonlinear, and first order or higher order. Partial differential equations can be classified as elliptic, parabolic, or hyperbolic, depending on their characteristics.

5. How do you solve a differential equation?

There are many methods for solving differential equations, including separation of variables, integration, substitution, and numerical methods. The most appropriate method to use depends on the type of differential equation and its complexity. In some cases, it may not be possible to find an exact solution, and numerical methods may be used to approximate the solution.

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