what is the differential equation of (6x+1)y^2 dy/dx + 3x^2 +2y^3=0
To find the differential equation for the given equation, we will follow these steps:
Step 1: Rearrange the equation to get the terms involving differentials on one side and all other terms on the other side.
=> (6x + 1)y^2 dy/dx = - (3x^2 + 2y^3)
Step 2: Divide both sides of the equation by (6x + 1)y^2 to isolate dy/dx.
=> dy/dx = - (3x^2 + 2y^3) / ((6x + 1)y^2)
Step 3: Simplify the expression by factoring out a common factor from the numerator.
=> dy/dx = - (3x^2 + 2y^3) / [(6x + 1)y^2]
=> dy/dx = - (3x^2 + 2y^3) / [y^2(6x + 1)]
Step 4: Simplify the expression further by canceling out the common factors, if any.
=> dy/dx = - (3x^2 + 2y^3) / y^2(6x + 1)
Hence, the differential equation for the given equation is: dy/dx = - (3x^2 + 2y^3) / y^2(6x + 1).