solve the separable differential equation. (use c for any needed constant.)\n\\frac{dy}{dx}=6xe^{-y}
solve the separable differential equation. (use c for any needed constant.)\n\\frac{dy}{dx}=6xe^{-y}
Answer
Explanation:
Step1: Separate variables
Separate the variables in the differential equation (\frac{dy}{dx}=6xe^{-y}). We can rewrite it as (e^{y}dy = 6x dx).
Step2: Integrate both sides
Integrate both sides of the equation. For the left - hand side, (\int e^{y}dy=e^{y}+C_1). For the right - hand side, (\int6x dx=6\times\frac{x^{2}}{2}+C_2 = 3x^{2}+C_2). So, (e^{y}=3x^{2}+C) (where (C = C_2 - C_1)).
Step3: Solve for (y)
Take the natural logarithm of both sides. (y=\ln(3x^{2}+C))
Answer:
(y = \ln(3x^{2}+C))