Differential Equation: xdy-ydx=sqrt(x^2+y^2)dx

T

Trasegar

New member
Joined
Oct 20, 2015
Messages
4
I have a simple basic differential equation exercise. xdy-ydx=sqrt(x^2+y^2)dx. I have tried to put dy on the left and dx on the right, getting xdy/ydx=sqrt[((x^2+y^2)/y) + 1]
I would like to know if it is correct, or if is there a better approach. After this separation of variables, I will do the substitution. Any suggestion above what
it should be.
thank you
 
Trasegar said:
I have a simple basic differential equation exercise. xdy-ydx=sqrt(x^2+y^2)dx. I have tried to put dy on the left and dx on the right, getting xdy/ydx=sqrt[((x^2+y^2)/y) + 1]
I would like to know if it is correct, or if is there a better approach. After this separation of variables, I will do the substitution. Any suggestion above what
it should be.
thank you
Click to expand...

First substitute:

u = y/x →du = (- ydx + xdy)/x^2............ edited

Transform your DE to:

du * x^2 = x * sqrt(1 +u^2) dx............ edited


Now continue....
 
Last edited by a moderator:
That is the problem - trying to solve DE by staring at the equation....

Pick up pencil and paper and work step-by-step......
 
stapel said:
Take a close look at the variables. ;)
Click to expand...

First I say U = y/x from here du = (Xdy - Ydx)/X^2
I can see where the - sign came from
Then I divide the original equation by X^2, to make the left side equal to du
From there I got du * X^2 = X * sqrt(1+u^2)dx. Again I can see where the - sign came from.

BTW my previous reply intend was to know why they express du = (Ydy - Xdy)/X^2
Thank you
 
Trasegar said:
First I say U = y/x from here du = (Xdy - Ydx)/X^2
I can see where the - sign came from
Then I divide the original equation by X^2, to make the left side equal to du
From there I got du * X^2 = X * sqrt(1+u^2)dx. Again I can see where the - sign came from.

BTW my previous reply intend was to know why they express du = (Ydy - Xdy)/X^2
Thank you
Click to expand...

Incorrect

du = (y*dx - x*dy)/x^2 ....( I had the sign wrong in my original post ... I corrected it)

It is derived from the quotient law of differentiation.
 
Solving the equation

Subhotosh Khan said:
That is the problem - trying to solve DE by staring at the equation....

Pick up pencil and paper and work step-by-step......
Click to expand...

After the substitution, I noticed there is du*x^2, so to put each differential with its correspondent variable, I moved things around and finally got
1/((1+U^2)^(1/2) du = (1/x)dx

2(1+U^2)^(1/2) - ln|x| + C

At this point, just go back to replace y/x = U

:(
 
You must log in or register to reply here.
Top