First Order Differential (Just identify)

In summary: For question 4, substitution is a good way to go about it. :smile:For question 5, try the integration first and then see if it works. :smile:
  • #1
Vathral
4
0
Not sure if I should have put this here or in the homework section but I am simply asking for direction on these problems.

I have 5 problems to work on and my biggest problem is identifying what method to use to find the answer. I don't need help answering because it's something I'd like to do myself but have some trouble with as I learn.


Don't know how to use the neat forum stuff yet but will learn as I become part of the community slowly.

Problems to work on...
http://img261.imageshack.us/img261/9066/matwk8.jpg [Broken]

Those are the problems and so far I can see...

#1 ?
#2 is a separable equation correct?
#3 ?
#4 is substitution and then exact?

Already marked that one exact since I did it already.
Can't tell which one would follow Bernoulli equation, integrating factor, linear first order.


Thank you to anyone who would kindly reply :x


Edit: Took a look at the rules again and see that this really should belong in the homework section. I'm very sorry, I'll ask an admin/mod to move it.
 
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  • #2
Welcome to PF!

Hi Vathral! Welcome to PF! :smile:

For #1, try the obvious substitution. :wink:
 
  • #3
[tex]x \frac{dy}{dx} - y = \frac{x^3}{y} e^y_x[/tex]

So
[tex]v = \frac{y}{x}[/tex]

[tex]y = vx[/tex]

[tex]\frac{1}{y} = \frac{x}{y}[/tex]

[tex]\frac{dy}{dx} = v+x \frac{dv}{dx}[/tex]



[tex]x [v + x\frac{dv}{dx}] - vx = \frac{x^2}{v} e^v[/tex]

[tex]xv + x^2 \frac{dv}{dx} - vx = \frac{x^2}{v} e^v[/tex]

[tex]x^2 \frac{dv}{dx} = \frac{x^2}{v} e^v[/tex]

[tex]\frac{dv}{dx} = \frac{e^v}{v}[/tex]

[tex]\frac{vdv}{e^v} = dx[/tex]

[tex]\frac{-v-1}{e^v} = x + c[/tex]

[tex] \frac{-\frac{y}{x}- 1)}{e^y_x} = x + c [/tex]

[tex] - \frac{y}{x} - 1 = (x + c) e^y_x[/tex]

[tex] -y - 1 * x^2 e^y_x + cxe^y_x[/tex]

[tex] y = x^2 e^y_x+cxe^y_x[/tex]

This is for #1 of course. Anyone see anything wrong with the final answer? Thank you for the identifying, would like to take these questions an extra step now.
 
  • #4
Vathral said:
[tex] - \frac{y}{x} - 1 = (x + c) e^y_x[/tex]

[tex] -y - 1 * x^2 e^y_x + cxe^y_x[/tex]

[tex] y = x^2 e^y_x+cxe^y_x[/tex]

Hi Vathral! :smile:

Fine up until that line :smile:

but what happened to the - 1 on the left? :redface:

For question 3, the only problem is the mixed terms (mixed x and y) …

so start by ignoring the 8x2dx, and try and turn the other two into an exact differential … and then put the 8x2dx back in, of course. :smile:
 

What is a first order differential equation?

A first order differential equation is an equation that involves an unknown function and its first derivative. It can be written in the form dy/dx = f(x,y), where y is the dependent variable and x is the independent variable. It is commonly used to model rates of change in various scientific and mathematical applications.

What is the difference between an ordinary and a partial first order differential equation?

An ordinary first order differential equation involves only one independent variable, while a partial first order differential equation involves multiple independent variables. Ordinary differential equations are typically used to describe single-variable systems, while partial differential equations are used for multi-variable systems.

What is the order of a first order differential equation?

The order of a differential equation is determined by the highest derivative present in the equation. Since a first order differential equation only involves the first derivative, it is considered a first order equation.

What are some real-life applications of first order differential equations?

First order differential equations are used in a variety of scientific and mathematical fields, such as physics, chemistry, biology, economics, and engineering. They can be used to model population growth, radioactive decay, heat transfer, chemical reactions, and many other phenomena.

What are initial value problems and how are they related to first order differential equations?

An initial value problem is a type of differential equation that involves finding the solution to an equation given an initial condition. This initial condition provides the starting point for solving the equation. First order differential equations are often used to model initial value problems and find solutions for various physical and mathematical systems.

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