Question

014. The differential equation (2xy+ y-tany) d x + (x*- xtan^2y +sec^2yldy = 0 is
a Exact
bl Non-Exact​

Answer 1

Answer:

I try to solve the second equation assuming that is as follows:

(te^xt +2x)dx/dt + xe^xt =0 or ((te^xt + 2x)dx +(xe^xt)dt =0.

This equation is exact because

(te^xt + 2x)_t = (xe^xt)_x = e^xt + txe^xt. Then the equation is a total differential

dF(x,y) =0 with solution F(x,y) = C obtained solving the system

F_x = te^xt + 2x

F_t = xe^xt . The solution to the first Eq. is F(x,t) = e^xt + x^2 + G(t). The function G(t)

is obtained from the second Eq.which gives G’(t) =0..Then G(t) = C and the solution is

F(x,t) = e^xt + x^2 = C .

Answer 2

Answer:

I try to solve the second equation assuming that is as follows:

(te^xt +2x)dx/dt + xe^xt =0 or ((te^xt + 2x)dx +(xe^xt)dt =0.

This equation is exact because

(te^xt + 2x)_t = (xe^xt)_x = e^xt + txe^xt. Then the equation is a total differential

dF(x,y) =0 with solution F(x,y) = C obtained solving the system

F_x = te^xt + 2x

F_t = xe^xt . The solution to the first Eq. is F(x,t) = e^xt + x^2 + G(t). The function G(t)

is obtained from the second Eq.which gives G’(t) =0..Then G(t) = C and the solution is

F(x,t) = e^xt + x^2 = C .