Master PDE Problem Solving: Separation of Variables Explained

[joshthekid]

Homework Statement

Solve y*∂Ψ/∂x-(x/3)∂Ψ/∂y

Homework Equations

The Attempt at a Solution

My teacher told me to try separation of variables but and I tried to set Ψ=X(x)Y(y) where X is a function of just X and Y is a function of just y but when I got the solution and put it into the original pde it did not work.

 

[blue_leaf77]

y*∂Ψ/∂x-(x/3)∂Ψ/∂y

Where is the equal sign?

My teacher told me to try separation of variables but and I tried to set Ψ=X(x)Y(y) where X is a function of just X and Y is a function of just y but when I got the solution and put it into the original pde it did not work.

Can you post your work as well as the answer you got?

 

[joshthekid]

Where is the equal sign?

Sorry the equal sign might help The equation is
y*∂Ψ/∂x-(x/3)*∂ψ/∂x=0

Can you post your work as well as the answer you got?

So first I defined Ψ(x,y)=X(x)Y(y)
thus the equation becomes
y*∂(X(x)Y(y)/∂x-(x/3)*∂(X(x)Y(y)/∂y=0
Rearranging and using the multiplication rule
y*Y(y)d(X(x))/dx=(x/3)*X(x)d(Y(y))/dy
Rearranging again
y*(1/X(x))d(X(x))dy=(x/3)*(1/Y(y))*d(Y(y))dx
then integrating
y^2/2*ln(X(x))=x^2/6*ln(Y(y))+c

That is as far as I got.

 

[SammyS]

Sorry the equal sign might help The equation is
y*∂Ψ/∂x-(x/3)*∂ψ/∂x=0

So first I defined Ψ(x,y)=X(x)Y(y)
thus the equation becomes
y*∂(X(x)Y(y)/∂x-(x/3)*∂(X(x)Y(y)/∂y=0

Rearrange the above so one side of the equation only depends on x and the other only on y .

Then each side must equal a constant, Right?

 

[HallsofIvy]

Sorry the equal sign might help The equation is
y*∂Ψ/∂x-(x/3)*∂ψ/∂x=0

So first I defined Ψ(x,y)=X(x)Y(y)
thus the equation becomes
y*∂(X(x)Y(y)/∂x-(x/3)*∂(X(x)Y(y)/∂y=0
Rearranging and using the multiplication rule
y*Y(y)d(X(x))/dx=(x/3)*X(x)d(Y(y))/dy
Rearranging again
y*(1/X(x))d(X(x))dy=(x/3)*(1/Y(y))*d(Y(y))dx
then integrating
y^2/2*ln(X(x))=x^2/6*ln(Y(y))+c

That is as far as I got.

That last step is incorrect- you cannot Integrate that way!
Instead, at the point where you have yX'/X= (x/3)Y'/Y, divide both sides by xy/3 to get 3X'/(xX)= Y'/(yY).
The left side depends only on x while the right side depends only on y. But the equation has to be true for all x and y. Imagine changing x while not changing y. Since y does not change the right side does not change. But that means the left side cannot change! That is 3X'/(xX)= C, a constant. Since 3X'/(xX)= Y'/(yY), we also have Y'/yY= C.

3X'/(xX)= C is the same as 3dX/dx= CxX, a separable differential equation.

Mod note: Removed some text as being too much help.