Question

(1 point) 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 differential, dF(, y) gives the differential equation. That is, level curves F(x,y) = C are solutions to the differential equation: dy 4x3 - y dx + 4y2 First rewrite as M(x,y) dic + N(x, y) dy = 0 where M(x,y) = and N(x,y) = If the equation is not exact, enter not exact, otherwise enter in F(, y) as the solution of the differential equation here = C.

Answer 1

For a differential equation of the form, max,y) da + M(x,y) dy = 0 co mont - ce nter then 0 in exact and its solution is given as; J ME,y) dx + J (N-1 y J ME . ) de) dy =( , then 0 is NOT exact. and if om cs Scanned with t # ) Oo C CamScanne

Consider the differential equation, 4a - da x + 4y2 >> (2 + xyl) dy = (4x2-y) de - (4x + y) da - (x = qyte) ay = 0 :: M(x,y) = 427-y Nfx, y) = + (x + 4yr) het (te cp) 01.4 C-fix+4yº)) - (70) 14 = 0 -1 = -1 is equation is exact.. Wow, we will write its solution. Scanned with CamScanner

- JM dx = S(tas-y) da (Note: We ignore integration constant in the intermediate steps, as we write constant c at the end) wowej Sodo (24-cy) - 0- x -LOC - - x Na M da = + (x+4y+) (-2) 3--494+ x = -4 ya Scanned with CamScanner

1. slow f Imde) y = S-opoly = - 443 - Solution is; - Mdx + ((N - y s de) dy = C 24-ay - fy :c + F12 g) = *- ay- $yoc XY - Scanned with CamScanner