Calc 2 differential equations

In summary, the conversation discusses the use of integrating factors in solving differential equations. The question is how the integrating factor, e^(x), disappears when multiplied to both sides. The answer is that the brackets must be multiplied out to show that the expressions are equivalent. This process is demonstrated through an example equation, and the use of integrating factors is further explained.
  • #1
nick.martinez
51
0
im in calculus 2 right now and we are doing differential equaitons right now. I am confused as to why when i find the integrating factor I(x)=e^(∫p(x) and when i multiply both sides i get
e^∫(p(x))[(dy/dx)+p(x)*y]=d(e^(p(x)dx)*y) how are they equal. i will give an example.

(dy/dx)+y=x*e^(x) the integrating factor is e^(∫dx)=e^(x) then i multiply both sides by
e^(x) which gives

e^(x)*[(dy/dx)+y]=e^(x)*[x*e^(x)]

which is equal to:

d[e^(x)*y]/dx=e^(x)*x; basically my question is how do you get here? how does e^(x)*dy/dx just disappear after i multiply the integrating factor to both sides?

please help with what happens after this step: e^(x)*[(dy/dx)+y]=e^(x)*[x*e^(x)]
 
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  • #2
Lets see... your example is: ##\renewcommand{\dyx}[1]{\frac{d #1}{dx}}##

$$\dyx{y}+y=xe^{x}$$ - integrating factor is: ##e^{\int dx}=e^x##
multiplying both sides by the integrating factor gives you:

$$e^x\left [ \dyx{y}+y \right ] = e^x\left [ xe^x \right ]$$ ... to from there to the end result - you must first multiply out the brackets.

But you can see that the two expressions are the same by doing the differentiation on the LHS:

$$\dyx{} [e^x y]$$

This is how you choose the integrating factor - you have to notice that this relation will work.
http://tutorial.math.lamar.edu/Classes/DE/Linear.aspx
 

1. What is the purpose of studying differential equations in Calc 2?

Differential equations are used to mathematically model and describe the behavior of systems that change over time. Understanding these equations allows scientists to make predictions and solve real-world problems in fields such as physics, engineering, and economics.

2. What are the main topics covered in Calc 2 differential equations?

The main topics covered in Calc 2 differential equations include first-order differential equations, higher-order differential equations, systems of differential equations, and applications of differential equations.

3. What are the key techniques used to solve differential equations in Calc 2?

The key techniques used to solve differential equations in Calc 2 include separation of variables, substitution, integrating factors, and using power series. These techniques allow us to find analytical solutions to differential equations.

4. Are there any real-world applications of differential equations that I should know about?

Yes, differential equations have numerous real-world applications. For example, they are used to model population growth, understand the spread of diseases, and predict the motion of objects in space. They are also used in engineering to design structures and optimize processes.

5. What are some tips for understanding and solving differential equations in Calc 2?

Some tips for understanding and solving differential equations in Calc 2 include practicing consistently, familiarizing yourself with different types of equations and their solutions, and understanding the underlying concepts and principles. It can also be helpful to work through examples and ask for clarification when needed.

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