exact differential equation

(x2-4xy-2y2)dx+(y2-4xy-2x2)dy=0

1. 👍 1
2. 👎 0
3. 👁 1,389
1. I assume you are looking for a solution?
We want a function F(x,y) such that

Fx(x,y) = x^2 - 4xy - 2y^2
and
Fy(x,y) = y^2 - 4xy - 2x^2

To make sure there is such an F, let
M = x^2 - 4xy - 2y^2
N = y^2 - 4xy - 2x^2

My = -4x - 4y
Nx = -4y - 4x

So, things look good.

Our solution is
F(x,y) = 1/2 x^3 - 2x^2y - 2xy^2 + 1/3 y^3 = C

1. 👍 2
2. 👎 1
2. Your differential equation is of the form
M(x,y) dx + N(x,y) dy = 0
with dN/dx = dM/dy (those being partial derivatives)
The last equation is a necessary and sufficient condition for a solution of the form
f(x,y) = 0
df/dx (partial) = x^2 -4xy -2y^2
and
df/dy (partial) = y^2 -4xy -2x^2

f(x,y) = x^3/3 -2yx^2 -2y^2x + g(y)
y^2 -4xy -2x^2 = -2x^2 -4yx -dg/dy
-dg/dy = y^2
g(y) = -y^3/3 + C

Thus the solution is
x^3/3 -2yx^2 -2y^2x -y^3/3 = C

1. 👍 1
2. 👎 0
3. Steve and I disagree on a couple of terms. He could be right. My math is rusty. Check the math yourself. There are some good tutorials on the "exact differential" method online.

1. 👍 0
2. 👎 0
4. Steve's first term should have read (1/3)x^3 while the last term should be positive (+(1/3)y^3).

It was probably a typo in each case, so there was not really a disagreement, especially everyone uses the same method of solution.

1. 👍 0
2. 👎 0
5. Finally answer will be x^3+y^3-6xy(x+y)=c

1. 👍 0
2. 👎 0

Similar Questions

1. AP Chemistry

Calculate ∆G0 for the following reaction. 4XY + 3Z2−→2Z3X2+ 2Y2 Substance ∆H0f S0kJ/mol J/mol·K XY 205.0 75.0 Y2 −82.0 37.0 Z2 −25.0 78.0 Z3X2 −245.0 137.0 Answer in units of kJ/mol

Can you help me solve for y? 8x-4xy=3 -2x-5y=10

3. Algebra 1

is this a linear equation 4xy + 2y = 9

4. calculus

Find dy/dx by implicit differentiation. arctan(2x^2y)=x+4xy^2

1. Differential Equation

what is the differential equation of (3x^2-2y^2)+(1-4xy)dy/dx=0

2. algebra 2

simplify:(4xy^2)^-3

3. Algebra

Simplify: (2x-3y4)3(x3 + y)0 / (4xy-2)3

1. math

(2x-3y4)3(x3 + y)0 / (4xy-2)3

2. algebra

What is the value of the expression 2x^2y / 4xy when x = 2 and y = 1? a) 1 b) 2 c) 4 d) 3 b

3. Calculus

Find the derivative if x^3 + 4xy^2 - y^4 = 7

4. Algebra

What is the factored form of the following expression? x2 – 4xy – 21y2