On this page we will look at another type of differential equations: linear differential equations in the form y' + P(x)y=Q(x) which can be solved by using an integrating factor. There are many applications of this type of differential equation. For example, in electrical engineering, we can solve Kirchhoff's Laws to model electrical circuits. To be successful with this topic, you will need to have strong foundations in the following areas: integration techniques (especially in the form \(\int\frac{f'(x)}{f(x)} \mathrm{d}x\)) and manipulation of logarithms and exponents. You can get some practice in these areas if you complete all the quizzes on this page before attempting the exam-style questions.
using the integrating factor \(I=e^{\int P(x)\mathrm{d}x}\)
Essentials
In order to use the integrating factor method, the differential equation needs to be in the form \(\large \frac{\text{d}y}{\text{d}x}+P(x)y=Q(x)\). In the following videos we look at how the method works.
Easier Example
In the following video we will look at an easier question that requires the integrating factor method to solve it. In particular, we will look at how the method works and why it works. You will see that you need to be confident in quite a few areas of the HL course to be able to carry it out. In this case, we need to use integration by recognition and integration by parts.
Find the general solution to the differential equation
In the following video we look at how we can find the particular solution to a differential equation using an integrating factor. This example is more challenging, since we have to re-arrange the differential equation to put it in the correct form and the integration is more difficult.
Find the particular solution to the differential equation
To solve differential equations with an integrating factor, we are often required to manipulate exponents of logarithmic functions. This will give you practice in that skill
Often differential equations with an integrating factor require us to carry out integrations in the form \(\large\int\frac{f'(x)}{f(x)} \mathrm{d}x=\ln |f(x)|+C\)
The following quiz wil help you practise these type of integrals
The integrating factor, \(I\)for the differential equation in the form \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\) can be found by evaluating \(\large I=e^{\int P(x) \mathrm{d}x}\)
The following quiz gives you some practice in finding the integrating factor
The integrating factor, \(I\)for the differential equation in the form \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\) can be found by evaluating \(\large I=e^{\int P(x) \mathrm{d}x}\)
What is the integrating factor for the differential equation \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+2y=e^x\)
\(I=e^{\int 2 \ \mathrm{d}x}\\ I=e^{2x}\)
The integrating factor, \(I\)for the differential equation in the form \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\) can be found by evaluating \(\large I=e^{\int P(x) \mathrm{d}x}\)
What is the integrating factor for the differential equation \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{y}{x}=\sin x\)
The integrating factor, \(I\)for the differential equation in the form \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\) can be found by evaluating \(\large I=e^{\int P(x) \mathrm{d}x}\)
What is the integrating factor for the differential equation \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+\frac{2}{x}y=x\)
The integrating factor, \(I\)for the differential equation in the form \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\) can be found by evaluating \(\large I=e^{\int P(x) \mathrm{d}x}\)
What is the integrating factor for the differential equation \(\large\frac{\mathrm{d}y}{\mathrm{d}x}-\frac{x^2y}{1+x^3}=x^2\)
The integrating factor, \(I\)for the differential equation in the form \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\) can be found by evaluating \(\large I=e^{\int P(x) \mathrm{d}x}\)
What is the integrating factor for the differential equation \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+y \tan x=\sin x\)
The integrating factor, \(I\)for the differential equation in the form \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\) can be found by evaluating \(\large I=e^{\int P(x) \mathrm{d}x}\)
What is the integrating factor for the differential equation \(\large x\frac{\mathrm{d}y}{\mathrm{d}x}-y=x^3\)
We need to put the differential equation in the correct form
The integrating factor, \(I\)for the differential equation in the form \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\) can be found by evaluating \(\large I=e^{\int P(x) \mathrm{d}x}\)
What is the integrating factor for the differential equation \(\large (1+x)\frac{\mathrm{d}y}{\mathrm{d}x}+y=1+x\)
We need to put the differential equation in the correct form
The integrating factor, \(I\)for the differential equation in the form \(\large\frac{\mathrm{d}y}{\mathrm{d}x}+P(x)y=Q(x)\) can be found by evaluating \(\large I=e^{\int P(x) \mathrm{d}x}\)
What is the integrating factor for the differential equation \(\large\sin x\frac{\mathrm{d}y}{\mathrm{d}x}+y \cos x=\mathrm{cosec} x\)
We need to put the differential equation in the correct form
On this page we will look at another type of differential equations: linear differential equations in the form y' + P(x)y=Q(x) which can be solved by using an integrating factor. There are many applications of this type of differential equation. For example, in electrical engineering, we can solve Kirchhoff's Laws to model electrical circuits. To be successful with this topic, you will need to have strong foundations in the following areas: integration techniques (especially in the form \(\int\frac{f'(x)}{f(x)} \mathrm{d}x\)) and manipulation of logarithms and exponents. You can get some practice in these areas if you complete all the quizzes on this page before attempting the exam-style questions.
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