Find the general solution of the differential equation. Solve for as a function of .

Question

Find the general solution of the differential equation.  Solve for  as a function of  .

Quizplus · Accepted Answer

The differential equation given can be written as:

dy/dx = -y - 1/x

This is a first-order linear differential equation in standard form:

dy/dx + P(x)y = Q(x)

Where P(x) = 1/x and Q(x) = -1/x.

The integrating factor for this equation is:

IF = e^∫P(x)dx
   = e^ln(x)
   = x

Multiplying both sides of the equation by the integrating factor, we get:

x(dy/dx) + xP(x)y = xQ(x)

Simplifying, we get:

d/dx(xy) = -1

Integrating both sides with respect to x, we get:

xy = -ln|x| + C

Dividing both sides by x, we get:

y = (-ln|x| + C)/x

Hence, the general solution of the differential equation is:

y = (-ln|x| + C)/x, where C is an arbitrary constant.

Therefore, the best choice is C.

Find the General Solution of the Differential Equation