solve the initial value problem by separation of variables dy/dx=-xy^2, y(1)=-0.25
To solve the initial value problem dy/dx = -xy^2 with the initial condition y(1) = -0.25, we can use the method of separation of variables.
The first step is to separate the variables by writing the equation as:
(1/y^2) dy = -x dx
Next, we integrate both sides of the equation with respect to their respective variables. The integral of (1/y^2) dy is:
∫(1/y^2) dy = ∫(-x) dx
This can be solved as:
(-1/y) = (-x^2)/2 + C
where C is the constant of integration.
Now, we can rearrange the equation to solve for y:
y = -1/((-x^2)/2 + C)
Next, we substitute the initial condition y(1) = -0.25 into the equation.
-0.25 = -1/((-1^2)/2 + C)
-0.25 = -1/(-1/2 + C)
To simplify further, we can apply the reciprocal:
-0.25 = -2/(1 - 2C)
Now, we can solve for the constant C by cross multiplication:
-0.25(1 - 2C) = -2
0.25 - 0.5C = -2
-0.5C = -2 - 0.25
-0.5C = -2.25
C = -2.25 / -0.5
C = 4.5
Now, substitute the value of C back into the equation for y:
y = -1/((-x^2)/2 + 4.5)
So, the solution to the initial value problem is y = -1/((-x^2)/2 + 4.5), with the initial condition y(1) = -0.25.