Separate the variables of the following differential equation.ysin2θdydθ=1ysin^2\theta\frac{dy}{d\theta}=1

y ∙ d y = c s c 2 θ∙dθ ⁡ y\bullet dy=csc^2\operatorname{\theta\bullet d\theta}

Separate the variables of the following differential equation. y s i n 2 θ d y d θ = 1 ysin^2\theta\frac{dy}{d\theta}=1

we're asked to separate the variables in the following differential equation, which is y times sine squared of theta times d y d theta equals one. Now here I have the variables y and theta, so that means that I want this y and d y together on the same side, and I want to move this sine squared theta and d theta so that they're together on the other. So the first thing that I'm going to go ahead and do here is divide both sides by that sine squared. When I do that, that will cancel over here on the left, just leaving me with y times dy d theta equals one over sine squared of theta. Now, in order to move this d theta over, I can go ahead and just multiply by d theta on both sides in order to get that to cancel and I'm left over here on the left side with y times d y. Then this is equal to one over sine squared of theta D theta. Now we can rewrite one over sine squared of theta because we know that this is just the cosecant squared. So rewriting this as cosecant squared theta times d theta and now we've successfully separated our variables. Let us know if you have any questions and I'll see you in the next one.

Separate the variables of the following differential equation. y s i n 2 ⁡ θ d y d θ = 1 ysin^2⁡\theta\frac{dy}{d\theta}=1

y ∙ d y = c s c 2 θ∙dθ ⁡ y\bullet dy=csc^2\operatorname{\theta\bullet d\theta}

y ∙ d y = s e c 2 θ∙dθ ⁡ y\bullet dy=sec^2\operatorname{\theta\bullet d\theta}

d y = c s c 2 θ∙dθ ⁡ dy=csc^2\operatorname{\theta\bullet d\theta}

d y = s e c 2 θ∙dθ ⁡ dy=sec^2\operatorname{\theta\bullet d\theta}

13. Intro to Differential Equations

Separable Differential Equations

Calculus

13. Intro to Differential Equations

Separable Differential Equations

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Multiple Choice

Separate the variables of the following differential equation.
$y s i n^{2} θ \frac{d y}{d θ} = 1$

$y ∙ d y = s e c^{2} θ∙dθ$

$y ∙ d y = c s c^{2} θ∙dθ$

$d y = c s c^{2} θ∙dθ$

$d y = s e c^{2} θ∙dθ$

Verified step by step guidance

Step 1: Start with the given differential equation: ysin²(θ) * (dy/dθ) = 1.

Step 2: Rewrite the equation to separate the variables. Multiply both sides by dθ and divide by ysin²(θ) to isolate dy and dθ: (1/y) * dy = (1/sin²(θ)) * dθ.

Step 3: Recognize that 1/sin²(θ) can be rewritten using the trigonometric identity for cosecant: 1/sin²(θ) = csc²(θ). Substitute this into the equation: (1/y) * dy = csc²(θ) * dθ.

Step 4: Multiply through by y to simplify the left-hand side: y * dy = csc²(θ) * dθ.

Step 5: The variables are now separated, with y terms on one side and θ terms on the other. The equation is ready for integration: ∫y * dy = ∫csc²(θ) * dθ.

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