TAM3B - Differential equations

 (x-y)dx-(x + y) dy =0

Exact Equation

If the first order partial derivatives of M(x,y) and N(x,y) are continuous then M dx + N dy = 0 is an exact equation if and only if 

∂M/∂y=∂N/∂x

Hence to solve the exact equation M dx + N dy = 0 

1. Integrate M with respect to x keeping y constant
2. Integrate those terms in N not containing x with respect to y
3. The sum of those two integrals equated to c is the solution.

EXAMPLE 1

(x^2-2xy+3y^2 )dx+(y^2+6xy-x^2 )dy=0

EXAMPLE 2

(2xy-sec^2⁡x )dx+(x^2+2y)dy=0

EXAMPLE 3

Find the value of ‘n’ for which
(x+ye^2xy )dx+nxe^2xy  dy=0   and solve it

EXAMPLE 4

(sin⁡x.tan⁡y+1)dx-(cos⁡x.sec^2⁡y )dy=0

EXAMPLE 5

(e^y+1)  cos⁡x  dx+e^y  sin⁡x  dy=0