Question

Give the general solution of dy/dx=2xy²-2x+3y²-3

Answer 1

To find the general solution of the differential equation dy/dx = 2xy² - 2x + 3y² - 3, we can use the method of separating variables and integrate both sides to find the general solution.

Step-by-step explanation:

To find the general solution of the differential equation dy/dx = 2xy² - 2x + 3y² - 3, we can use the method of separating variables.

1. Move all terms containing dy to one side of the equation: dy/(3y² - 2y) = (2x - 3)dx.
2. Separate the variables by multiplying both sides by dx and dividing both sides by (3y² - 2y): (1/(3y² - 2y))dy = (2x - 3)dx.
3. Integrate both sides with respect to their respective variables: ∫(1/(3y² - 2y))dy = ∫(2x - 3)dx.
4. Solve the integrals to find the general solution.

The integration will involve techniques such as partial fractions and u-substitution, which should be done step by step and may lead to different forms of the solution depending on the values of the constants involved.