Understanding Integrating Factors for Calculus Exams

[Hyperreality]

Just a question to clear things up regarding to the integrating factors for my exams tomorrow:

The standard first order differential equation has the form:

dy/dx + p(x)*y = q(x).

where the integrating factor is e^(integral of p(x) dx).

But in one of the previous year paper, it has the equation

dy/dt + bt = a

where "b" and "a" are constants. To me, this does not resembles the standard form.

But apparently it is, in the answer they used the integral factor
e^(integral of b) and solved the equation.

I must have missed some important points in how to identify integrating factors...

 

[cepheid]

It's a special case of the standard from in which p(x) and q(x) are constant functions! So, it's an easier to solve case. But it's still a linear equation, so why wouldn't the method of integrating factors work? The standard form is just the most general case.

Edit: Why did you post a question about differential equations in the Kindergarten to Grade 12 HW Help forum?

Edit: I didn't read the problem carefully before. Furthermore, it should be even easiser to solve, because p(x) = 0! There is no 'y' term.

Edit: At third glance, the problem is SO easy, that although it is a linear equation, it doesn't even merit the method of integrating factors! You have dy/dt = q(t). Why not just integrate directly?

$$\frac{dy}{dt} = a - bt$$

$$y = \int{(a-bt)}dt$$

Right?

 

[wais]

Thank you for sharing your question and concerns regarding integrating factors for your calculus exams. It is important to have a clear understanding of integrating factors in order to successfully solve differential equations.

Firstly, it is important to note that the standard form of a first order differential equation is dy/dx + p(x)*y = q(x). This is the form in which we can identify and use the integrating factor e^(integral of p(x) dx). However, as you have observed, there are cases where the equation may appear different, such as in the example you provided with dy/dt + bt = a.

In this case, it may not seem like the standard form, but it can still be solved using the same method. The key is to look for a common factor between the terms in the equation. In this case, the common factor is "t". So, we can rewrite the equation as (1/t)*dy/dt + b*(1/t)*y = a/t. This is now in the standard form, where p(x) = 1/t and q(x) = a/t. Therefore, the integrating factor is e^(integral of 1/t dt) = e^(ln|t|) = t.

It is also important to note that the integrating factor can be any function that satisfies the condition e^(integral of p(x) dx). In the example you provided, the answer may have used e^(integral of b) because it is a constant and satisfies the condition.

In summary, the key to identifying the integrating factor is to look for a common factor between the terms in the equation and use that to rewrite the equation in the standard form. I hope this clears things up and helps you in your exam tomorrow. Good luck!