# Differential Equation- Separation of Variables

In summary, the conversation discusses a problem involving the derivative of y with respect to x, which can be solved by using a u-substitution. However, there is some confusion about the correct u-substitution to use. The solution involves rewriting the sec function in terms of cosine and using u=x^2 and du=2x.dx.

## Homework Statement

y'=xsec^2(x^2)

2. The attempt at a solution
dy/dx=xsec^2(x^2)
dy=xsec^2(x^2)dx
$$\int$$dy=$$\int$$xsec^2(x^2)dx
lny= (here i'll do a u substitution)
----
u=x^2 du=1/3x^3dx

... and here's my problem. It seems like that creates a very difficult u-sub to try and manage. Any suggestions/help?

lny= (here i'll do a u substitution)
----
u=x^2 du=1/3x^3dx
This is incorrect. Your method is correct, though.

I think you should write sec function in terms of cos. Function will be dy = x / (cos^2(x^2)). Then if you say x^2 = u and du = 2x.dx, you can solve the problem easily.

u=x^2 du=2xdx
2$$\int$$du/cos^2(u)

You're off by a factor of 1/4. It should be $$\frac{1}{2} \int sec^2u \ du$$.

## What is the concept behind separation of variables in differential equations?

The concept behind separation of variables in differential equations is to split a complex equation into simpler equations that can be solved individually. This is done by separating the dependent and independent variables and putting them on opposite sides of the equation.

## How do you separate variables in a differential equation?

To separate variables in a differential equation, you must move all terms containing the dependent variable to one side and all terms containing the independent variable to the other side. This allows you to integrate each side separately.

## What are the steps for solving a differential equation using separation of variables?

The steps for solving a differential equation using separation of variables are: 1) separate the variables, 2) integrate each side, 3) solve for the constant of integration, if necessary, and 4) combine the solutions to get the final solution.

## What are the limitations of separation of variables in solving differential equations?

Separation of variables can only be used for certain types of differential equations, specifically those that are separable. It also cannot be used for differential equations with more than two variables or with higher-order derivatives.

## What are some real-world applications of differential equations that use separation of variables?

Differential equations and separation of variables are used in many fields of science and engineering, including physics, chemistry, economics, and biology. Some specific applications include modeling population growth, predicting chemical reactions, and analyzing electrical circuits.