17calculus 17calculus
First Order Second Order Laplace Transforms Additional Topics Applications, Practice
Separation of Variables
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Integrating Factors (Linear)
Substitution
Exact Equations
Integrating Factors (Exact)
Linear
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Substitution
Reduction of Order
Undetermined Coefficients
Variation of Parameters
Polynomial Coefficients
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Chebyshev Equations
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Solve Initial Value Problems
Classify Differential Equations
Fourier Series
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Practice Problems
Practice Exam List
Exam A1
Exam A3
Exam B2

You CAN Ace Differential Equations

17calculus > differential equations > exact equations

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Exact Differential Equations and Integrating Factors

classification: this technique applies to first-order differential equations

\( M(x,y)~dx + N(x,y)~dy = 0 \) and \(M_y = N_x\)

Exact differential equations are first-order differential equations of the form \(\displaystyle{ M(x,y)~dx + N(x,y)~dy = 0 }\) where \(M_y = N_x\). The requirement that \(M_y = N_x\) means the differential equation is what we call exact. Here are some equivalent ways of writing this differential equation.

\(\displaystyle{ M(x,y)~dx + N(x,y)~dy = 0 }\)

\(\displaystyle{ M(x,y) + N(x,y)y' = 0 }\)

\(\displaystyle{ M(x,y) + N(x,y)\frac{dy}{dx} = 0 }\)

How To Solve

Lets jump right into a video to see how we solve these. Here is a great video clip explaining the technique and showing a couple of examples.

Dr Chris Tisdell - exact equations

In the video above, he does a good job of emphasizing that, before you start actually solving the differential equation, you need to make sure it is exact by calculating \(M_y\) and \(N_x\) and checking that they are equal. Otherwise, this technique will take you down a path that leads nowhere. Most instructors will take off points if you don't show work that you did this. So check with your instructor. But even if they don't take off points, it could save you a lot of time on homework and exams if you do this and it takes only a few seconds. So it is a good investment of your time.

The main difference with these types of problems is that you need to remember the constant of integration is actually a function of integration. This occurs because we are taking the partial integral of one variable while holding the other variable constant. This concept is covered in detail on the multi-variable partial integration page.

Here is another video that is very similar to the first one, with a few different examples. But watching this one also, will help you understand this technique better.

Dr Chris Tisdell - more exact equations

Integrating Factors

If we have a first-order equation of the form \( M(x,y)~dx + N(x,y)~dy = 0 \) but \(M_y \neq N_x\), i.e. the equation is not exact (also called inexact), we may be able to convert the equation to exact using integrating factors. You first came across integrating factors when you studied linear, first-order equations. This technique also uses an integrating factor but it is calculated differently.

There are actually two possible integrating factors that convert the differential equation to an exact equation.

\(\displaystyle{ \mu_1 = \exp \int{ \frac{M_y - N_x}{N} dx } }\)

 

\(\displaystyle{ \mu_2 = \exp \int{ \frac{M_y - N_x}{-M} dy } }\)

The trick comes in when you are asked to evaluate these integrals. The evaluation is not always possible and can get quite messy. Also, one integral may yield an integrating factor that is quite complicated while the other one may be much easier to use. Regardless, these integrating factors can, at times, be quite useful. Let's look at an example in this next video.

Khan Academy - Integrating Factors

Okay, time for some practice problems.

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Practice Problems

Instructions - - Unless otherwise instructed, solve these differential equations. Make sure to check that the equation is exact before attempting to solve. Give your answers in exact form.

Level A - Basic

Practice A01

\(2xy~dx+(x^2+3y^2)~dy=0\)

solution

Practice A02

\(2xy~dx+x^2~dy=0\)

solution

Practice A03

\(2x+3+(2y-2)y'=0\)

answer

solution

Practice A04

\(2xy~dx+(x^2-1)~dy=0\)

answer

solution

Practice A05

\([\cos(x)\sin(x)-xy^2]~dx+y(1-x^2)~dy=0\)

answer

solution

Practice A06

\((y\cos(x)+2xe^y)+(\sin(x)+x^2e^y-1)y'=0\)

answer

solution

Practice A07

\( (2xy+1) + (x^2+3y^2)y'=0 \)

solution

Practice A08

Find the explicit solution for the exact differential equation \( (e^{x+y}-2x)dx + (e^{x+y}+2y)dy = 0\), \(y(0)=0\).

answer

solution


Level B - Intermediate

Practice B01

\((3x+y+1)~dx+(3y+x+1)~dy=0\)

solution

Practice B02

\((2x+y+1)~dx+(2y+x+1)~dy=0\)

solution

Practice B03

\((3x^2-2xy+2)~dx+(6y^2-x^2+3)~dy=0\)

solution

Practice B04

\( (2xy^{-1}+2ye^{2x}-1) +\) \( (e^{2x}+y^2-x^2y^{-2})\frac{dy}{dx}=0 \)

solution

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