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Running Head: DIFFERENTIAL EQUATION

Differential Equation

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1

DIFFERENTIAL EQUATION

2

Question 1

Consider the differential equation

(2x − 2y 2 )dx + (12y 2 − 4xy)dy = 0

a) Show that the differential equation is exact

Let M(x, y) = (2x − 2y2 )

And

N(x, y) = (12y2 − 4xy)

d

My (x, y) = dy M(x, y) = −4y

Nx (x, y) =

d

N(x, y) = −4y

dx

The differential equation are exact because Nx (x, y) = Nx (x, y) = −4y

hence the differential equation is exact

b) Solve the differential equation

We now integrate Y with respect to x to find the function 𝑓(𝑥, 𝑦) such that 𝑑𝐹 = 0 up to yindeterminacy in F.

F(x, y) = ∫(2x − 2y2 )dx = x 2 − 2xy2 + g(y)

d

Then Fy (x, y) = dy F(x, y) = −4xy + g ′ (y) which is also N(x, y) = (12𝑦2 − 4𝑥𝑦)

Therefore g ′ (y) = 12𝑦2

Then g(y) = ∫ 12y2 dy = 4y3 + c

Therefore the solution of the Differential equation is

𝐅(𝐱, 𝐲) = 𝐱 𝟐 − 𝟐𝐱𝒚𝟐 + 𝟒𝒚𝟑 + 𝐜

DIFFERENTIAL EQUATION

3

Question 2

Consider the differential equation

(2ycosx + xex )dx + (2sinx + 3y 2 )dy = 0

a) Show that the ...

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