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}

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0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

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.
$y s i n^{2} θ \frac{d y}{d θ} = 1$

$y ∙ d y = s e c^{2} θ∙dθ$

$y ∙ d y = c s c^{2} θ∙dθ$

$d y = c s c^{2} θ∙dθ$

$d y = s e c^{2} θ∙dθ$

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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.