Solving Exact Differential Equations

Here I introduce you to a method for solving the first-order differential equation

M(x, y)dx + N(x, y)dy = 0

for the special case in which this equation represents the exact differential of a function z = f(x, y).

step 1: Test for Exactness

Let M and N have continuous partial derivatives. The differential equation is exact if and only if ­­­

∂­­M/∂­­y = ∂­­­N/∂­­­x.

step 2: 

equation 1 -> ∂­­u/∂­­x = M                          equation 2 -> ∂­­­u/∂­­y = N

step 3: Integrate equation 1 with respect to x

equation A -> int ∂u = int M ∂­­x + k(y)

Note: If you integrate equation 1 then take constant as k(y), as above. Otherwise if you integrate equation 2, which is with respect to y, then you’ll have to take h(x) as a constant. 

step 4: Partially differentiate with respect to y

∂­­­u/∂­­y = ∂­­­/∂­­y  int M ∂­­x + ∂­­­/∂­­y k(y)

use the equation 1 and 2 from the step 2 to substitute,

∂­­­u = M ∂­­x,                M ∂­­x/∂­­y = N.

So, left and right side cancels,

∴ ∂­­­/∂­­y k(y) = 0,       k(y) = 0.

step 5: Put the value of k(y) or h(x) in equation A & finally substitute u=c.

 

Example Question: (x3 + 3xy2)dx + (3x2y + y2)dy = 0

M = (x3 + 3xy2),       N = (3x2y + y2).

Differential w.r.t y -> My = 6xy,

Differential w.r.t x -> Nx = 6xy.

step 1: My = Nx (Exact).

step 2: equation 1 -> ∂­­u/∂­­x = (x3 + 3xy2),                    equation 2 -> ∂­­­u/∂­­y = (3x2y + y2).

step 3: Integrate equation 1 w.r.t x     int ∂u = int (x3 + 3xy2) ∂­­x

equation A -> u = x4/4 + 3/2 x2y2 + k(y).

step 4: Partially differentiate equation A w.r.t y

N = ∂­­­u/∂­­y = 3x2y+ ∂­­­/∂­­y k(y)

now substitute N,

3x2y + y2 = 3x2y+ ∂­­­/∂­­y k(y)

y2 = ∂­­­/∂­­y k(y)

integrate to get the value of k(y),

k(y) = y3/3.

step 5: substitute k(y) and u=c,

x4/4 + 3/2 x2y2 + y3/3 = c.

 

1 Comment
  1. Reply
    Kyla Pozar February 6, 2018 at 12:53 AM

    The abacus has change into precious teaching device in colleges,
    tuition centers and neighborhood centers, and is used by residence education dad and mom around the globe
    Understanding mathematics learning objects based on the speculation of learning
    Gagne.

    Leave a reply

    BrilliantInfo
    Login/Register access is temporary disabled