Question

Solve the following initial value problem for the function y(x).
y' = 2xy² y(1) = 1/2

Answer 1

The initial value problem y' = 2xy² with y(1) = 1/2 is solved using separation of variables and integrating, giving us the particular solution y = -1/(x² - 3) which satisfies the given condition.

Step-by-step explanation:

We have been given a differential equation y' = 2xy² with the initial condition y(1) = 1/2. To solve this initial value problem, we can use the method of separation of variables.

First, rearrange the differential equation to separate the variables y and x:

1/y² dy = 2x dx

Now, integrate both sides:

Integral of 1/y² dy = integral of 2x dx

After integration, we get:

-1/y = x² + C

Next, apply the initial condition y(1) = 1/2 to solve for C.

-2 = 1 + C

=> C = -3

So the particular solution of the differential equation is:

y = -1/(x² - 3)

This equation satisfies both the differential equation and the initial condition given in the problem.