Question

use separation of variables to find the general solution of the differential equation 6 6(ln(y))y ′−x 3y=0. (Use symbolic notation and fractions where needed. Use C as an arbitrary constant. Absorb into C as much as possible. First, give the smallest answer, then the largest.)

Answer 1

To find the general solution of the differential equation, use separation of variables and integrate both sides. The general solution is y = e^[(1/6)(1/2)x^2 + C/6].

Step-by-step explanation:

To find the general solution of the given differential equation, we can use separation of variables. Rearrange the equation to separate the variables:

6(ln(y))y ′ = x 3y

Divide both sides by y:

6(ln(y))y ′/y = x 3

Integrate both sides with respect to x:

∫[6(ln(y))y ′/y] dx = ∫(x 3) dx

Using the properties of natural logarithms, we can simplify the left side:

∫6(ln(y)) d(ln(y)) = ∫(x 3) dx

Integrate both sides:

6(ln(y)) + C1 = 1/2x^2 + C2

Subtract C1 from both sides:

6(ln(y)) = 1/2x^2 + C2 - C1

Combine C2 - C1 into a single constant, C:

6(ln(y)) = 1/2x^2 + C

Divide both sides by 6:

ln(y) = (1/6)(1/2)x^2 + C/6

Take the exponential of both sides:

y = e^[(1/6)(1/2)x^2 + C/6]

This is the general solution to the differential equation.