What is the solution to the following differential equation?

Question

Quizplus · Accepted Answer

The differential equation 11ead0bc_8bec_eb4c_99a0_85453b3f30f2_TB6988_11 is a separable equation, which means we can separate the variables and integrate. Starting with:

dy/dx = (x^3 + y^3)/(3xy^2)

We can rearrange to get all the y's on one side and all the x's on the other:

(3y^2)dy = (x^3+y^3)dx

Next, we integrate both sides:

∫(3y^2)dy = ∫ (x^3+y^3)dx

Simplifying, we get:

y^3 = (1/4)x^4 + (1/2)y^2 + C

where C is the constant of integration. Solving for y:

y = [1/2 ± sqrt((x^4/16) + C - (1/4)(x^2))]^(1/2)

We can simplify this further as:

y = [1/2 ± sqrt((x^4+4C-x^2)/16)]^(1/2)

Therefore, the solution is given by choice D: 11ead0bc_8bed_3970_99a0_65e710fff826_TB6988_11.

What Is the Solution to the Following Differential Equation