Differential Equations
Differential Equations
Allen Star Batch
Grade 12
Question:
Given the differential equation $\frac{dy}{dx} = \frac{6x^2}{2y + \cos y}$, $y(1) = \pi$ and the following statements
Solution is $y^2 - \sin y = 2x^3 + c$
Solution is $y^2 + \sin y = 2x^3 + c$
$c = \pi^2 - 2$
$c = \pi^2 + 2$
Step-by-Step Solution
Key Concept: This is a separable differential equation solved by integrating both sides with respect to their respective variables.
Given $\frac{dy}{dx} = \frac{6x^2}{2y + \cos y}$, separate variables to get $(2y + \cos y)dy = 6x^2 dx$. Integrate both sides: $\int(2y + \cos y)dy = \int 6x^2 dx$ yields $y^2 + \sin y = 2x^3 + c$. Using the condition $c = \pi^2 - 2$, the solution is $y^2 + \sin y = 2x^3 + \pi^2 - 2$.
Correct Answer: 2,3