Question

Diff. eq question. Please see below

Answer 1

Find the value of b for which the equation
\[(xy+ ^{2} bx ^{2} y) dx + (x + y)x ^{2} dy = 0\]
is exact, and then solve it using that value of b.

Answer 2

I had similar problem solved, but then the professor changed it to this problem, now it's a little tougher :(

Answer 3

Well this equation is exact if there is a function F = F(x,y) such that the differential you have written above satisfies
\[\frac{\partial F}{\partial x} \ dx + \frac{\partial F}{\partial y} \ dy = 0\]

Answer 4

So you have an equation for Fx and Fy (where this means the partial derivatives of F wrt x and y respectively).

Solve them and make the F you find consistent.

Answer 5

so i need deriv in respect to x any y first, then figure out a function that would make the two derivs = ??

Answer 6

For example if Fy = (x+y)x^2, then integrating wrt y, we have

F(x,y) = x^3.y + (xy)^2/2 + X(x)

where X(x) is some arbitrary function in x.

Do the same thing with the expression for Fx and then set the two expressions for F equal to each other and solve.

Answer 7

James, how did you type that notation, I didn't see it in the eq editor

Answer 8

int(M(x,y)); add g(y); derive down to N(x,y) is what i recall it as

Answer 9

right click the equation and show source

Answer 10

gotcha, lemme give it a shot on paper and I'll be back in a sec

Answer 11

\[\frac{\partial F}{\partial x} \ dx + \frac{\partial F}{\partial y} \ dy = 0\]

\frac{\partial F}{\partial x} \ dx + \frac{\partial F}{\partial y} \ dy = 0

Answer 12

ahhh nice, thanks

Answer 13

the place it between the delimiters:\[\text{\[\]}\] there just aint no good what to type up an example that is usable

Answer 14

what? lol

Answer 15

you say ain't ?

Answer 16

i say isnt and type aint , its a war me any fingers are having

Answer 17

wow this is weird, now this site says I'm someone else, "tarekitani". Do you guys see this as well?