Final answer:
To find the general solution, we need to integrate both sides of the equation. Integrating the left side with respect to y, we get ln|y| + C₁ = (4/3)x³ + 9x + C.
Step-by-step explanation:
The given differential equation is y' = 4yx² + 9. To find the general solution, we need to integrate both sides of the equation.
Integrating the left side with respect to y, we get ∫(1/y)dy = ∫4x² + 9 dx.
Using the power rule of integration, the left side simplifies to ln|y| + C₁, where C₁ is the constant of integration.
For the right side, we integrate term by term. The integral of 4x² is (4/3)x³ + C₂, and the integral of 9 is 9x + C₃. Combining these results, we have ln|y| + C₁ = (4/3)x³ + 9x + C₂ + C₃.
Combining constants, we can simplify further and write the general solution as ln|y| = (4/3)x³ + 9x + C, where C is the constant of integration that combines C₁, C₂, and C₃.