Question:

Consider the differential equation:

(b) Find the particular solution that satisfies the initial condition $y(1) = 2$.

Answer:

(a) To solve the given differential equation using separation of variables, we start by rewriting the equation in the form:

Rearranging the equation, we have:

Next, we separate the variables by integrating both sides of the equation:

Integrating, we get:

where C is the constant of integration. Thus, the general solution to the differential equation is $y = x^3 + C$.

(b) Using the initial condition $y(1) = 2$, we can find the particular solution by substituting the values into the general solution:

Simplifying, we find:

Subtracting 1 from both sides, we obtain:

Therefore, the particular solution that satisfies the initial condition is $y = x^3 + 1$.

In summary, the solution to the given differential equation using the method of separation of variables is $y = x^3 + C$, where C is the constant of integration. The particular solution that satisfies the initial condition $y(1) = 2$ is $y = x^3 + 1$.