Final answer:
To solve the differential equation dy/dx = 6x²y², we can separate the variables and integrate. The general solution is y = -1/(2x³ + C), where C is the constant of integration. By substituting an initial condition, we can find the particular solution.
Step-by-step explanation:
To solve the differential equation dy/dx = 6x²y², we can separate the variables and integrate. Start by dividing both sides by y²: (1/y²)dy/dx = 6x². Then integrate both sides with respect to x. On the left side, integrate (1/y²)dy as -1/y, and on the right side, integrate 6x² as 2x³. So we have -1/y = 2x³ + C, where C is the constant of integration.
To find the solution explicitly, we can rearrange the equation to solve for y: y = -1/(2x³ + C). This is the general solution to the differential equation.
For example, if we have an initial condition y(0) = 1, we can substitute this into the general solution to find the particular solution. -1/(2(0)³ + C) = 1. Solving for C, we get C = -1/2. Now substitute C = -1/2 back into the general solution, we have y = -1/(2x³ - 1/2). This is the particular solution to the given differential equation with the initial condition y(0) = 1.