Question

can anyone explain to be the differential equations: first order (exact equations) ? for example
(3x^2y^2-4xy)y' + 2xy^3-2y^2=0

Answer 1

first...turn y' into dy/dx
\[\implies (3x^2 y^2 - 4xy)\frac{dy}{dx} + 2xy^3 - 2y^2 = 0\]
now...multiply dx to all sides
\[\implies (3x^2 y^2 - 4xy) dy + (2xy^2 - 2y^2)dx = 0\]
following so far?

Answer 2

yea

Answer 3

then what happens?

Answer 4

now you see if it's exact

Answer 5

how?

Answer 6

\[\frac{\partial M}{\partial y} = \frac{\partial N}{\partial x}\]
for \[Mdx + Ndy = 0\]
in this case.. your M = (2xy^3 - 2y^2)
and N = 3x^2 y^2- 4xy

so \[\frac{\partial M}{\partial y} = 6xy^2 - 4y\]
\[\frac{\partial N}{\partial x} = 6xy^2 - 4y\]
so it's exact

got it?

Answer 7

i see so what happens after that? do i just say it is exact because i think my teacher wants be to show that it is a implicit solution or something

Answer 8

you solve it now... take the \[\int Mdx\]
so in this case..
\[\implies \int (2xy^3 - 2y^2)dx\]
\[\implies\int 2xy^3 dx - \int 2y^2 dx\]
\[\implies x^2y^3 - 2y^2 x + g(y)\]
(that g(y) is kind of like the arbitrary constant c)

now derive it WITH RESPECT TO Y
\[\implies 3xy^2 y^2 - 4y^2 x + g^\prime(y)\]
still following?

Answer 9

yes !!!
omg!

Answer 10

uhh wait that should be \[\implies 3x^2 y^2 - 4xy + g^\prime (y)\]
lots of typos

Answer 11

oh okay lol

Answer 12

now equate this to N so..
\[\implies 3x^2 y^2 - 4xy + g^\prime (y) = 3x^2 y^2 - 4xy\]
cancel the cancellable...
\[\implies g^\prime (y) = 0\]
now integrate both sides
\[\int g^\prime (y) = \int 0\]
\[\implies g(y) = 0\]
substitute this into the g(y) written earlier...

\[\implies x^2 y^3 - 2y^2 x + g(y)\]
if you substitute
\[\implies x^2 y ^3 - 2y^2 x + 0\]
\[\implies x^2 y^3 - 2y^2 x\]
this is now the final answer

Answer 13

oh i see! so integrate the Mdx and then equal to N and see if they are equal
wow this was a great lesson! thank you so much!

Answer 14

great work Lgba

Answer 15

welcome

Answer 16

and thanks @UnkleRhaukus

Answer 17

wow i think i should learn more from you im actually taking differential equation & linear algebra this semester and i think it would really help out if you keep updating my questions =) Thank You so much!