First Order Linear Differential Equation - I can't solve it

In summary, the conversation is about solving a differential equation dx/dt = 0.63 - (9x / 2060). One approach is to find the integrating factor, which is e^(9t/2060), and multiply it by both sides of the equation. Integrating the left side results in 144.2e^(9x/2060), but the right side needs to be in terms of x, so it is divided by e^(9t/2060). However, this approach may not make sense in the context of the question. The other approach is to separate the equation by finding an equivalent form of dx/(0.63 - (9x / 2060))=dt, and then
  • #1
jumbogala
423
4

Homework Statement


Solve the differential equation dx/dt = 0.63 - (9x / 2060).

Homework Equations


The Attempt at a Solution


I started by finding the integrating factor. So I integrated 9/2060, to get 9t/2060. Therefore e^(9t/2060) is my integrating factor.

Multiply 0.63 by that integrating factor to get 0.63e^(9t/2060) on the left side of the equation.

Integrate the left side, and you get 144.2e^(9x/2060). The right side of the equation looks like (e^(9t/2060))x right now.

I need this in a function of x so I divide by e^(9t/2060)... and I get x = 144.2.

But that doesn't make sense in the context of the question. What I have should be something that varies with t...
 
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  • #2
You don't need an integrating factor. It's separable. You just have to integrate something of the form dx/(a+bx)=dt. It's just a log.
 
  • #3
Oh... shouldn't it work the way I did it anyway though? I thought I could always use an integrating factor if I wanted to...

I can't figure out how to separate it =/
 
  • #4
jumbogala said:
Oh... shouldn't it work the way I did it anyway though? I thought I could always use an integrating factor if I wanted to...

I can't figure out how to separate it =/

You can use an integrating factor if you do it half way carefully, sure. But where did t come from?? And how did it disappear at the end? You were being sloppy.

dx/(0.63 - (9x / 2060))=dt. There. It's separated. Now just integrate it.
 

FAQ: First Order Linear Differential Equation - I can't solve it

What is a first order linear differential equation?

A first order linear differential equation is an equation that involves a function and its derivative, where the highest power of the derivative is 1. It can be written in the form y' + P(x)y = Q(x), where P(x) and Q(x) are functions of x.

Why is it difficult to solve first order linear differential equations?

First order linear differential equations can be difficult to solve because they do not have a general formula for finding a solution. Instead, each equation must be solved using specific techniques and methods, and the solutions can vary depending on the form of the equation.

What are some common techniques for solving first order linear differential equations?

Some common techniques for solving first order linear differential equations include separation of variables, integrating factors, and substitution methods. These techniques involve manipulating the equation in various ways to isolate the dependent and independent variables and find a solution.

What are the initial conditions in a first order linear differential equation?

The initial conditions in a first order linear differential equation refer to the values of the dependent and independent variables at a specific point. These conditions are typically given in the form of y(x0) = y0, where x0 is the initial value of the independent variable and y0 is the initial value of the dependent variable.

What are some tips for solving first order linear differential equations?

Some tips for solving first order linear differential equations include carefully identifying the type of equation and selecting an appropriate method, paying attention to the initial conditions, and double-checking the solution by plugging it back into the original equation. It is also helpful to practice solving various types of first order linear differential equations to become more familiar with the process.

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