Question:

A differential equation that can be solved using the method of separation of variables is given by:

(1 + y^2) dx - y dy = 0

(a) Use separation of variables to solve the given differential equation.

(b) Find the particular solution to the initial condition y(1) = 0.

Answer:

(a) To solve the given differential equation using separation of variables, we will separate the variables x and y on different sides of the equation and integrate both sides. The steps are as follows:

(Step 1) Rearrange the equation to separate x and y:

(1 + y^2) dx - y dy = 0

dx = (y dy) / (1 + y^2)

(Step 2) Integrate both sides with respect to their respective variables:

∫ dx = ∫ (y dy) / (1 + y^2)

(Step 3) On the left side, the integral ∫ dx represents the integral of a constant, which gives us x:

x = ∫ dy / (1 + y^2)

(Step 4) On the right side, we can use a substitution to simplify the integral. Let u = y^2 + 1, then du = 2y dy:

x = ∫ (1/2) du / u

(Step 5) Evaluate the integral on the right side:

x = (1/2) ln|u| + C

(Step 6) Substitute u = y^2 + 1 back into the equation:

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

Thus, the general solution to the given differential equation is:

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

(b) To find the particular solution that satisfies the initial condition y(1) = 0, we substitute x = 1 and y = 0 into the general solution:

1 = (1/2) ln|(0)^2 + 1| + C

Simplifying the expression:

1 = (1/2) ln|1| + C

1 = (1/2) (0) + C

C = 1

Therefore, the particular solution to the initial condition y(1) = 0 is:

x = (1/2) ln|y^2 + 1| + 1

This completes the solution to the given differential equation.