Question

Which of the following differential equation is exact?

A. (y2+ x)dx+ (y3 + x)dy = 0
B. (x + y) (dx - dy) = dx + dy
C. 2xydx + (x + y2) dy = 0
D. None of the above

Answer 1

B

Explanation:

Solve (x + y(x)) (-(dy(x))/(dx) + 1) = (dy(x))/(dx) + 1:

Let v(x) = x + y(x), which gives (dv(x))/(dx) = (dy(x))/(dx) + 1:

(-(dv(x))/(dx) + 2) v(x) = (dv(x))/(dx)

Simplify:

-((dv(x))/(dx) - 2) v(x) = (dv(x))/(dx)

Solve for (dv(x))/(dx):

(dv(x))/(dx) = (2 v(x))/(v(x) + 1)

Divide both sides by v(x)/(v(x) + 1):

((dv(x))/(dx) (v(x) + 1))/v(x) = 2

Integrate both sides with respect to x:

integral((dv(x))/(dx) (v(x) + 1))/v(x) dx = integral2dx

Evaluate the integrals:

log(v(x)) + v(x) = 2 x + c_1, where c_1 is an arbitrary constant.

Solve for v(x):

v(x) = W(e^(2 x + c_1))

Simplify the arbitrary constants:

v(x) = W(c_1 e^(2 x))

Substitute back for y(x) = -x + v(x):

Answer: y(x) = -x + W(c_1 e^(2 x))