Question

Solve the differential equation, such that the equation passes through the given point ( x, y). (Remember to use absolute values where appropriate.) dy/dx = x - 28/x, (-1, 6)

a) y = x² - 28 ln|x|
b) y = x² + 28 ln|x|
c) y = x² + 28 ln(-x)
d) y = x² - 28 ln(-x)

Answer 1

The differential equation dy/dx = x - 28/x is solved through integration, yielding the general solution y = x^2/2 - 28 ln|x| + C. Using the initial condition (-1, 6), we find C = 11/2, leading to y = x^2/2 - 28 ln|x| + 11/2. The closest answer option, without the constant term, is y = x^2 - 28 ln|x|.

Step-by-step explanation:

To solve the differential equation dy/dx = x - 28/x and find the particular solution that passes through the given point (-1, 6), we integrate the equation with respect to x.

The antiderivative of x is x2/2, and the antiderivative of -28/x is -28 ln|x|. Therefore, the general solution of the differential equation is:

y = x2/2 - 28 ln|x| + C, where C is the constant of integration.

Now we use the initial condition (-1, 6) to find C:

6 = (1/2) (-1)2 - 28 ln|-1| + C

6 = 1/2 - 28*(0) + C

Therefore, C = 6 - 1/2 = 11/2.

Inserting the value of C into the general solution gives us:

y = x2/2 - 28 ln|x| + 11/2

Since we are looking for an answer without the constant term (which would be combined in the options given), the closest answer is.

a) y = x2 - 28 ln|x|