how do you solve the initial value problem by using separation of variables dy/dx=-xy^2 when y(1)=-0.25

To solve the initial value problem dy/dx = -xy^2 with the initial condition y(1) = -0.25 using separation of variables, we need to employ the following steps:

Step 1: Rearrange the equation to separate the variables.
The given differential equation is dy/dx = -xy^2. We can rearrange it as: (1/y^2)dy = -x dx.

Step 2: Integrate both sides of the equation.
Integrate both sides with respect to their respective variables. The left side integrates to ∫(1/y^2)dy, and the right side integrates to ∫-x dx.

∫(1/y^2)dy = -∫x dx

Step 3: Evaluate the integrals.
To integrate ∫(1/y^2)dy, we can use the power rule of integration, which states that ∫x^n dx = (1/(n+1))x^(n+1) + C.

∫(1/y^2)dy = -∫x dx
-1/y = -x^2/2 + C,

where C is the constant of integration.

Step 4: Solve for y.
To solve for y, we need to isolate it. Multiply both sides of the equation by -1: 1/y = x^2/2 - C.

Taking the reciprocal of each side yields: y = 1/(x^2/2 - C).

Step 5: Apply the initial condition.
Now, we can substitute the given initial condition y(1) = -0.25 into the equation and solve for C.

y(1) = 1/(1^2/2 - C)
-0.25 = 1/(1/2 - C)
-0.25 = 2/(1-2C)

To simplify further, multiply both sides by (1-2C):
-0.25(1-2C) = 2

Expand and solve for C:
-0.25 + 0.5C = 2
0.5C = 2.25
C = 4.5

Step 6: Substitute the value of C into the equation obtained in Step 4.
Now, substitute C = 4.5 into y = 1/(x^2/2 - C):

y = 1/(x^2/2 - 4.5)

So, the solution to the initial value problem dy/dx = -xy^2, y(1) = -0.25 is y = 1/(x^2/2 - 4.5).