Final answer:
To solve the differential equation dy/dx = x² - 5/y with the given initial condition, variables are separated, integrated, and the initial condition is used to find the constant of integration, resulting in the particular solution f(x) = √[2(⅓ x³ - 5x - 9/27)].
Step-by-step explanation:
To find the particular solution to the differential equation d y/d x = x² - 5/y with the initial condition f(-3) = 2, we first separate variables and integrate both sides:
- Multiply both sides by y to get y dy = (x² - 5) dx.
- Integrate both sides: ∫ y dy = ∫ (x² - 5) dx, yielding ½ y² = ⅓ x³ - 5x + C, where C is the constant of integration.
- Apply the initial condition to solve for C: ½ (2)² = ⅓ (-3)³ - 5(-3) + C gives us C = ½ - ⅓ - 15 = ½ - 9 - 15 = -⅓.
- So the particular solution is y = ±√[2(⅓ x³ - 5x - ⅓)].
However, since the initial condition is f(-3) = 2, we choose the positive square root because y is positive. Hence, the particular solution is f(x) = √[2(⅓ x³ - 5x - ⅓)].