Final answer:
To solve the initial value problem y' = 6y² + xy² with the initial condition y(0) = 1, we need to find the solution y(x) and determine where it attains its minimum value. We can solve this problem using separation of variables.
Step-by-step explanation:
To solve the initial value problem y' = 6y² + xy² with the initial condition y(0) = 1, we need to find the solution y(x) and determine where it attains its minimum value. We can solve this problem using separation of variables.
- Separate the variables by moving all terms with y to one side and all terms with x to the other side: y²dy = (6y² + xy²)dx
- Integrate both sides: ∫ y²dy = ∫ (6y² + xy²)dx
- Integrate the left side: (1/3)y³ + C1 = ∫ (6y² + xy²)dx
- Integrate the right side: (1/3)y³ + C1 = 2y³ + (1/2)xy³ + C2
- Combine like terms: (1/3)y³ - 2y³ = (1/2)xy³ + C2 - C1
- Simplify the equation: -(5/3)y³ = (1/2)xy³ + C3
This is the solution to the initial value problem. To determine where the solution attains its minimum value, we can examine the behavior of the function y(x). However, since the equation is nonlinear, it may not be easy to find the minimum analytically. To find the minimum numerically, we can use methods such as graphing or optimization algorithms.