Use the mixed partials check to see if the following differe

Use the \"mixed partials\" check to see if the following differential equation is exact. If it is exact find a function F(x, y) whose level curves are solutions to the differential equation (2e^x sin(y) + 2y)dx + (2x + 2e^x cos(y))dy = 0 F(x, y) =

Solution

Use the mixed partials check to see if the following differential equation is exact. If it is exact find a function F(x,y) whose level curves are solutions to differential equation

M(x,y)= 2e^x sin y + 2y
N(x,y) = 2x + 2e^x cos y

del M/del y = 2e^x cos y + 2
del N/del x = 2e^x cos y + 2
As both are equal, So, its exact.
F(x,y) = integrate N(x,y) dy + h(x)
= integrate 2x + 2e^x cosy dy + h(x)
= 2xy + 2e^x sin y + h(x)
del F/ del x = M
So, 2x + 2e^x sin y + h\'(x) = 2e^x sin y + 2y
So, h\'(x) = 0
h(x) = 0
So solution is

F(x,y) = 2xy + 2e^x sin y