Question

Solve the given initial-value problem. The differential equation is homogeneous. Find the solution to xy² dy/dx = y³ - x³, given that y(1) = 3.

Answer 1

The initial-value problem was solved using separation of variables and the initial condition y(1) = 3. By integrating both sides and applying the initial condition, we determined the solution to the differential equation to be y = (2 ln|x| - 9)⁻½, considering the positive branch for x > 0.

Step-by-step explanation:

To solve the initial-value problem where the differential equation is homogeneous with the form xy² dy/dx = y³ - x³, and given that y(1) = 3, we can use the method of separation of variables.

First, we rearrange the equation:

dy/y³ = dx/x dy/(y - x(y/x)³).

We separate variables and integrate both sides:

∫ dy/y³ = ∫ dx/x.

Solving the integrals, we get

-1/2 y² = ln|x| + C,

where C is the integration constant. To find C, we use the initial condition y(1) = 3:

-1/2 (3)² = ln|1| + C,

C = -1/2 (9).

Finally, the solution to the differential equation is:

y = ±(2 ln|x| - 9)⁻½.

Considering only the positive branch (since y(1) = 3>0) we get the explicit solution for y:

y = (2 ln|x| - 9)⁻½ where x > 0 since y(1) = 3.