Ugly first order differential equation

In summary, an ugly first order differential equation is a type of mathematical equation that involves a function and its first derivative. It is difficult to solve analytically and often requires numerical methods. These equations typically have non-linear terms and cannot be solved using standard techniques. Some real-world applications include modeling population growth and chemical reactions. To solve these equations, one can use numerical methods such as Euler's method or consult with resources such as online tutorials or tutors for assistance.
  • #1
cameuth
17
0

Homework Statement


solve the differential equation:
(1+t^2)y'+4ty=(1+t^2)^-2

Homework Equations



μ=exp∫adt

The Attempt at a Solution


this problem gets quite ugly, so here goes.
first question
does μ=e^(1+t^2)^2
 
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  • #2
I don't think the integrating factor is as bad as you think.
write the equation as [itex] y'+ \frac{4ty}{1+t^2}= \frac{1}{(1+t^2)^3} [/itex]
now what do I need to multiply this equation through to run the product rule backwards on the left hand side.
 
  • #3
cameuth said:
μ=e^(1+t^2)^2
Nope. How did you get that?
 

FAQ: Ugly first order differential equation

What is an ugly first order differential equation?

An ugly first order differential equation is a type of mathematical equation that involves a function and its first derivative. It is considered "ugly" because it is difficult to solve analytically and often requires numerical methods.

How can I identify an ugly first order differential equation?

Ugly first order differential equations typically have non-linear terms, such as trigonometric or exponential functions, and may also contain variables raised to non-integer powers. They can be identified by the fact that they cannot be solved using standard techniques like separation of variables.

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

Ugly first order differential equations are commonly used in physics, engineering, and other scientific fields to model complex systems and phenomena. Examples include population growth, chemical reactions, and electrical circuits.

How do I solve an ugly first order differential equation?

To solve an ugly first order differential equation, you can use numerical methods such as Euler's method, Runge-Kutta methods, or the finite difference method. These methods approximate the solution by breaking down the equation into smaller parts and using iterative calculations.

Are there any resources available to help me understand and solve ugly first order differential equations?

Yes, there are many online resources, textbooks, and software programs available to help you understand and solve ugly first order differential equations. You can also consult with a mathematics or engineering tutor for additional guidance and support.

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