What is the general solution to the differential equation dydx=8e4x2⁢cos(2y)?

sin ( 2 y ) = 4 ∫ e 4 x 2 d x + C

What is the general solution to the differential equation d y d x = 8 e 4 x 2 ⁢ cos ( 2 y ) ?

sin ( 2 y ) = 4 ∫ e 4 x 2 d x + C

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13. Intro to Differential Equations

Basics of Differential Equations

Calculus

13. Intro to Differential Equations

Basics of Differential Equations

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Multiple Choice

What is the general solution to the differential equation $\frac{d y}{d x} = 8 e^{4 x^{2}} cos (2 y)$ ?

sin (2 y) = 4 \int e^{4 x^{2}} d x + C

y = 4 x^{2} cos (2 y) + C

y = 8 e^{4 x^{2}} + C

tan (y) = 8 e^{4 x^{2}} + C

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Verified step by step guidance

Step 1: Recognize that the given differential equation is separable, meaning it can be written in the form f(y)dy = g(x)dx. Start by rewriting the equation: $ \frac{dy}{dx} = 8e^{4x^2} \cos(2y) $.

Step 2: Separate the variables y and x. Divide both sides by $ \cos(2y) $ and multiply by dx to isolate terms involving y on one side and terms involving x on the other: $ \frac{1}{\cos(2y)} dy = 8e^{4x^2} dx $.

Step 3: Simplify the left-hand side using the trigonometric identity $ \frac{1}{\cos(2y)} = \sec(2y) $. The equation becomes $ \sec(2y) dy = 8e^{4x^2} dx $.

Step 4: Integrate both sides. On the left-hand side, integrate $ \sec(2y) dy $, which involves a standard trigonometric integral. On the right-hand side, integrate $ 8e^{4x^2} dx $, which may require substitution or other techniques depending on the form of $ e^{4x^2} $.

Step 5: After integration, introduce the constant of integration $ C $ and express the solution in terms of the relationship between y and x. The general solution will involve the integral $ \int e^{4x^2} dx $ and the trigonometric function $ \sin(2y) $, leading to a form like $ \sin(2y) = 4 \int e^{4x^2} dx + C $.