dy/dx = 4xy^2
I need to find the integral of that. So I separate the variables and have 1/y^2 dy = 4x dx. Where do I go from here?
y^-2 dy = 4 x dx
-1 y^-1 + c = 2 x^2 the important step
-1/y + c = 2 x^2
-1 + c y = 2 x^2 y
y (2 x^2 -c) = -1
y = -1/ (2 x^2 -c) you can write that several ways
To integrate the equation, you correctly separated the variables by rearranging the equation as 1/y^2 dy = 4x dx.
Next, you can integrate both sides of the equation separately.
On the left-hand side, integrate 1/y^2 dy with respect to y. This involves using the power rule for integration, which states that ∫x^n dx = (x^(n+1))/(n+1), where n is not equal to -1.
In this case, n is equal to -2, so applying the power rule, we have:
∫1/y^2 dy = ∫y^(-2) dy = y^(-2+1)/(-2+1) = -y^(-1)/1 = -1/y.
On the right-hand side, integrate 4x dx with respect to x. This involves using the power rule again, this time with n equal to 1:
∫4x dx = 4 ∫x^1 dx = 4(x^(1+1))/(1+1) = 4x^2/2 = 2x^2.
Now, after integrating both sides, the equation becomes:
-1/y = 2x^2 + C,
where C is the constant of integration.
Finally, you have obtained the integral of dy/dx = 4xy^2 as -1/y = 2x^2 + C.
To find the integral of the given equation, you have already separated the variables and obtained 1/y^2 dy = 4x dx. Now, you can integrate both sides of the equation with respect to their respective variables.
Integrating the left side:
∫(1/y^2) dy = ∫dy/y^2
Using the power rule of integration, the integral of y^n with respect to y is (y^(n+1))/(n+1). Here, n = -2, so:
∫(1/y^2) dy = (-1/y^1+1)/(1+1) = -1/2y.
Integrating the right side:
∫4x dx = 4∫x dx
Using the power rule of integration, the integral of x^n with respect to x is (x^(n+1))/(n+1). Here, n = 1, so:
∫4x dx = 4(x^(1+1))/(1+1) = 2x^2.
Now, you can equate the results of the integrations:
-1/(2y) = 2x^2.
To solve for y, you can multiply both sides by -1/2:
1/(2y) = -2x^2.
To isolate y, take the reciprocal of both sides:
2y = -1/(2x^2).
Divide both sides by 2:
y = -1/(4x^2).
Therefore, the integral of dy/dx = 4xy^2 is y = -1/(4x^2).