Understanding Differential Equations and their Solutions

In summary, the conversation discusses two examples of differential equations and their solutions using the technique of separation of variables. The first example has a general solution of y= x+\frac{x^3}{3} + c, while the second example has a general solution of y= tan(x+c). The conversation also mentions the importance of integrating both sides correctly.
  • #1
beetle2
111
0

Homework Statement



Hi Guys,
I've got a couple of examples of de's in my lecture notes. one is:
[itex]y'=1+x^2[/itex]
which has a general solution of [itex]y= x+\frac{x^3}{3} + c[/itex]
which i I understand they have taken the integral wrt [itex]x[/itex]


the second is

[itex]y'=1+y^2[/itex] which has a general solution of [itex]y= [/itex]tan[itex](x+c)[/itex]

Can some one please explain to me how that got the second solution?
 
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  • #2
beetle2 said:

Homework Statement



Hi Guys,
I've got a couple of examples of de's in my lecture notes. one is:
[itex]y'=1+x^2[/itex]
which has a general solution of [itex]y= x+\frac{x^3}{3} + c[/itex]
which i I understand they have taken the integral wrt [itex]x[/itex]


the second is

[itex]y'=1+y^2[/itex] which has a general solution of [itex]y= [/itex]tan[itex](x+c)[/itex]

Can some one please explain to me how that got the second solution?
Both problems can be done by the technique of separation of variables.

For the first, you have
dy/dx = 1 + x^2
==> dy = (1 + x^2)dx
Integrate both sides to get what you already have shown.

For the second, you have
dy/dx = 1 + y^2
==> dy/(1 + y^2) = dx
Now integrate both sides to get
arctan(y) = x + C ==> y = tan(x + C).
 
  • #3
Thanks mate I had forgoten about separation of variables.
I was integrating both sides wrt x in the second example.
 
  • #4
beetle2 said:
Thanks mate I had forgoten about separation of variables.
I was integrating both sides wrt x in the second example.
And good luck with that!:biggrin:
 

1. What is a differential question?

A differential question is a type of question that asks for a comparison or contrast between two or more variables or concepts. It is often used in scientific research to explore relationships and differences between different groups or conditions.

2. How is a differential question different from other types of questions?

A differential question is unique in that it specifically focuses on the comparison or contrast between variables, rather than just asking for information about a single variable. It also often requires the use of statistical analysis to answer.

3. What are some examples of differential questions?

Examples of differential questions include: "Is there a significant difference in test scores between students who received extra tutoring and those who did not?", "Are there differences in brain activity between individuals with and without anxiety disorders?", and "How does the effectiveness of two different medications compare in treating a specific condition?"

4. How do you formulate a good differential question?

To formulate a good differential question, you should first identify the variables or concepts that you want to compare. Then, consider what specific aspect of these variables you are interested in comparing (such as differences in means, proportions, or correlations). Finally, make sure the question is clear, specific, and answerable with the available data or methods.

5. What are the benefits of using differential questions in scientific research?

Differential questions allow researchers to uncover relationships and differences between variables, which can provide valuable insights and help advance scientific knowledge. They also allow for more precise and specific hypotheses to be tested, leading to more reliable and accurate results.

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