dy/dx=(6x^2 *y^2)
dy/dx=(6x^2 *y^2)
dy/y^2 = 6x^2 dx
-1/y = 2x^3 + c
y = -1/(2x^3+c)
or
y = 1/(c-2x^3) (different c)
why 1/y? that the only thing I don't get.
huh? did you forget the power rule?
∫ 1/y^2 dy = -1/y
To solve the differential equation dy/dx = 6x^2 * y^2, we can use the method of separable variables. Here's how you can do it:
Step 1: Rearrange the equation
Start by rearranging the equation so that all terms involving y and dy are on one side, and all terms involving x and dx are on the other side:
dy/y^2 = 6x^2 dx
Step 2: Separate the variables
Now, separate the variables by dividing both sides of the equation by y^2 and multiplying both sides by dx:
(1/y^2) dy = 6x^2 dx
Step 3: Integrate both sides
Let's integrate both sides of the equation with respect to their respective variables. Remember to include the integration constant (C) on the right-hand side.
∫(1/y^2) dy = ∫6x^2 dx
This simplifies to:
-1/y = 2x^3 + C
Step 4: Solve for y
Next, we solve the equation for y. Begin by multiplying both sides of the equation by -y to isolate 1/y:
-1 = -2x^3y + Cy
Rearrange the equation to put it in a more standard form:
2x^3y - Cy = 1
At this point, the equation is in implicit form and can be difficult to solve explicitly for y. However, if you have an initial condition (such as y(x0) = y0), you can use it to find the particular solution by substituting the given values of x0 and y0 into the equation.
Please note that the solution provided above assumes that the integration has been carried out correctly. It's always a good practice to double-check the calculations done during each step to ensure their accuracy.