how do you separate the variable dy/dx=x^2/y given y=-5 when x=3
To separate the variables in the equation dy/dx = x^2/y, you need to rearrange the terms involving x and y on opposite sides of the equation. Here's how you can do it:
1. Start with the given equation: dy/dx = x^2/y.
2. Multiply both sides of the equation by y to get rid of the fraction: y * dy/dx = x^2.
3. Rearrange the equation by bringing all the terms involving y to one side: y * dy = x^2 * dx.
Now that we have separated the variables, we can integrate both sides of the equation to find the solution.
4. Integrate both sides with respect to their respective variables:
∫ y * dy = ∫ x^2 * dx.
To integrate y * dy, we treat y as the variable and dy as the differential. Integrating y * dy gives us (1/2)*y^2 + C1, where C1 is the constant of integration.
To integrate x^2 * dx, we treat x as the variable and dx as the differential. Integrating x^2 * dx gives us (1/3)*x^3 + C2, where C2 is another constant of integration.
5. The equation now becomes: (1/2)*y^2 + C1 = (1/3)*x^3 + C2.
Note that C1 and C2 are arbitrary constants that can be determined if initial conditions or boundary conditions are given.
Given y = -5 and x = 3, we can substitute these values into the equation and solve for the constants:
(1/2)*(-5)^2 + C1 = (1/3)*(3)^3 + C2.
25/2 + C1 = 27/3 + C2.
25/2 + C1 = 9 + C2.
From here, it is not possible to determine the specific values of C1 and C2 without additional information. However, you can simplify the equation further and express it in terms of y and x.