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### Course: Differential equations>Unit 1

Lesson 7: Exact equations and integrating factors

# Exact equations example 1

First example of solving an exact differential equation. Created by Sal Khan.

## Want to join the conversation?

• I still don't understand how (psi)(x,y) can equal C, it was equal to y*sin(x)+x^2*e^y-y+C, right? How does that statement equal C exactely?
• The whole premise of the solution is that you have an equation that you can represent as

M(x) + N(y) = 0

You DEFINE dψ/dx = M(x) and dψ/dy = N(y). Then the above equation becomes

dψ/dx + dψ/dy = 0

or, in other words,

ψ'(x,y) = 0

Integrate both sides and you end up with ψ(x,y) = C.
• What happened to the yprime that was multiplied by N in the first line of the video?
• From Sbeecroft's 3 year old reaction: "1. if an expression is of the form M(x,y) + N(x,y)[dy/dx], and"

because of the y' he recognized a certain form of the equation, so tried to see if the equation indeed also agreed to the two other requirements for it to be in the "exact differential equation"-form:

2. if M(x,y) is the partial differential of some function Ѱ with respect to x, and;
3. if N(x,y) is the partial differential of some function Ѱ with respect to y;
• Everything is fine, but at the end when we write the final solution my instructor writes it as ψ(x,y,c) NOT ψ(x,y) . as in this case it would be ψ (x,y,c)= y sinx + x^2 e^y - y+ C
And he once emphasized on writing that c
• C is just a constant. It is not an independent variable for which you can plug different values into ψ. So, there is no real reason for writing ψ(x,y,c).
• At , why isn't f(y) = -y plus some function of x?
• It can't have x in it by definition. Because f(y) was defined as a sole function of y @ , as it was referring to 'constant' value of the partial integral of psi with respect to x.
• At , I don't understand why the differential equation can be rewritten as:

d/dx psi(x,y) = 0

Do we always rewrite this as d/dx psi(x,y) = 0? Is this something we do just to find c? I'm just not seeing where this is coming from.
• The first two videos in this chain are critical to making that jump. Check out video "Exact equations intuition 2" at about to recap: the right-hand side of the chain rule equation for d/dx psi(x,y) at the top and the left-hand side of the differential equation at the bottom are essentially identical, leading to this equation via substitution.
(1 vote)
• 1：42。。。what's he means cosx is a constan...why doesn't he use product rule?
(1 vote)
• when you take partial derivative respect to only one variable you treat the other varibale as constant so since he is taking partial respect to y, he treats all finctions of x as constant. Hope it makes sence now :)
• at how is the anti-derivative of f'(y)=1 is f(y)=y+c instead of f(y)=c?
• The antiderivative of a constant is a polynomial of the first order, remember that the power rule when integrating adds one power to the variable, and a constant is like having a variable to the `0` power, so after integrating you would get the variable to the `1` power. That is why the `1` becomes `y`.
• At , it is a little error in video when he writes M_y=cosx+1xe^y ( it should be M_y=cosx+2xe^y).
(1 vote)
• No, he wrote it correctly. It's just a low resolution video, like many of the "old" KA videos are.