## anonymous 3 years ago (2x^2+xy+y^2)dx + 2x^2 dy=0 How Do I Solve This 1st Order DifEQ? Help Please!!

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1. anonymous

yup this 1st Order DifEQ

2. anonymous

you can use y=vx (for homogeneous equations) to solve it

3. anonymous

I don't understand how you determine if its homogeneous, separable...but I understand if its an exact equation...

4. anonymous

then dy =xdv +vdx

5. anonymous

I have 6 problems that i have to do and i have the answers but i just dont understand how to do the work to get the answer unless its an exact equation..

6. anonymous

see the sum of the powers of x and y is uniform (here its 2 ) throughout hence homogeneous

7. anonymous

Great! I'm Learning Something Finally!

8. anonymous

welcome

9. anonymous

after using y=vx and dy =xdv +vdx the resultant will become a seperable one

10. anonymous

Is there a certain formula that is out there where i can solve these problems if they are separable or homogenous or any other thing out there?? I really want to learn this stuff

11. abb0t

You cannot solve this using separation of variables.

12. anonymous

Yes your right, I understand that its not an exact equation..

13. anonymous

I have 6 problems that are due in 45 mins... I need help!!!

14. JamesJ

Let's start from the beginning (2x^2+xy+y^2)dx + 2x^2 dy=0 implies that dy/dx = -(2x^2+xy+y^2)/(2x^2) and hence dy/dx = -1 - (1/2)(y/x) - (y/x)^2

15. JamesJ

This is what is called a homogeneous equation. It is one of the meanings of the word homogeneous for differential equations. The substitution 'trick' for such equations is to write v = y/x and hence y = vx which implies y' = v + xv' Therefore we can rewrite the equation above as v + xv' = - (1 + (1/2)v + v^2) This now IS a separable equation which you can solve using standard techniques to find the function v(x). Once you have done that, substitute back v(x) = y(x)/x to solve for y(x)

16. JamesJ

***I dropped a half in front of the (y/x)^2 term, so v + xv' = -(1/2) ( 2 + v + v^2 )

17. JamesJ

Make sense?

18. anonymous

I keep getting different answers though when i do that... i have the solutions just can get there while showing my work

19. anonymous

Anyone who can help solve these problems will be a life saver!

20. JamesJ

Well, v + xv' = -(1/2) ( 2 + v + v^2 ) implies -2x v' = v^2 + 3v + 2 = (v+2)(v+1) hence separating $-2\int \left( \frac{A}{v+2} + \frac{B}{v+1} \right) dv = \int \frac{dx}{x}$ You take it from there.