Partial Derivative Equation as solutions

Nov 2019
2
0
Odense
I have to solve PDE as ODE, but my lecturer didn't give me enough help and tutorial, to solve these problems.
I searched YouTube and other websites to find a clue how to solve them but I find no clue. Please help!

The problems are:

i have to solve PDE as ODE where \(\displaystyle u = u(x,y)\)

\(\displaystyle u_y + u = e^{xy}\)
\(\displaystyle u_{xx} = 4y^2u\)
\(\displaystyle u_y = 2*x*y*u\)

I already know about separation of variables, and terms...
 

romsek

MHF Helper
Nov 2013
6,665
3,002
California
There must be some typo here. By subtracting the first equation from the third you are able to solve for $u(x,y)= \dfrac{e^{x y}}{2 x y+1}$
by simple algebra. But that solution does not satisfy the second equation.
 
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topsquark

Forum Staff
Jan 2006
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Wellsville, NY
I had figured this to be three problems, not one?

They don't separate but I can't believe that your instructor would just give you a set of problems that haven't (and won't) be discussed either in class or the reading.

Skim the chapter and see if you find this: Method of Characteristics.

-Dan

Addundum:

Actually, the second two equations do separate. Just remember than when integrating over, say x, we take y to be a constant.
For example, in the second one we have
Let u(x, y) = X(x)Y(y)
\(\displaystyle u_{xx} = 4y^2u \implies X''Y = 4y^2 XY \implies \dfrac{X''}{X} = 4y^2 \text{ and } Y \neq 0\)
You would treat the y^2 as a constant.

Since these last two are separable and you haven't gone further than that(?) is it possible that the first has a typo? Possibly \(\displaystyle u_y + u = e^{x + y}\) ? This does separate.

-Dan
 
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Nov 2019
2
0
Odense
I had figured this to be three problems, not one?

They don't separate but I can't believe that your instructor would just give you a set of problems that haven't (and won't) be discussed either in class or the reading.

Skim the chapter and see if you find this: Method of Characteristics.

-Dan

Addundum:

Actually, the second two equations do separate. Just remember than when integrating over, say x, we take y to be a constant.
For example, in the second one we have
Let u(x, y) = X(x)Y(y)
\(\displaystyle u_{xx} = 4y^2u \implies X''Y = 4y^2 XY \implies \dfrac{X''}{X} = 4y^2 \text{ and } Y \neq 0\)
You would treat the y^2 as a constant.

Since these last two are separable and you haven't gone further than that(?) is it possible that the first has a typo? Possibly \(\displaystyle u_y + u = e^{x + y}\) ? This does separate.

-Dan
I know how to seperate, but the first and third don't, how do i solve it?
 

topsquark

Forum Staff
Jan 2006
11,568
3,453
Wellsville, NY
I know how to seperate, but the first and third don't, how do i solve it?
The third one separates, too.

Again, let u(x,y) = X(x)Y(y)
\(\displaystyle u_y = 2xyu \implies XY'= 2xyXY\)

So
\(\displaystyle \dfrac{1}{y} \dfrac{Y'}{Y} = 2x\) so long as \(\displaystyle X \neq 0\).

As to the first check out the link. It's pretty much step by step.

-Dan