Differential Calculus Problems With Solution

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Calculus problem. Help with separating differential equations?

Suppose dy/dθ = y^5cosθsin^5θ.

1. Separate the differential equation, then integrate both sides.
2. Write the general solution as a function y(θ).

Help is much appreciated. Thanks!

dy / dθ = y^5 cosθ sin^5θ

dy = y^5 cosθ sin^5θ dθ

(1/y^5) dy = cosθ sin^5θ dθ

That is the separation.

Let’s integrate ∫cosθ sin^5θ dθ = ∫sin^5θ cosθ dθ
You will have to use integration by parts.

u = sin^5θ
du = 5(sin^4θ)cosθ dθ

dv = cosθ dθ
v = sinθ

∫u dv = uv – ∫v du

∫sin^5θ cosθ dθ = (sin^5θ)(sin^θ) – ∫sinθ*5(sin^4θ)*cosθ dθ
∫(sin^5θ)(cosθ) dθ = (sin^6θ) – 5∫(sin^5θ)(cosθ) dθ

The left side and right side have a common integral so add 5∫(sin^5θ)(cosθ) dθ to both sides.

6∫(sin^5θ)(cosθ) dθ = (sin^6θ)

Divide both sides by 6.

∫(sin^5θ)(cosθ) dθ = [(sin^6θ)/6] + C

Let’s integrate
∫(1/y^5) dy = ∫(y^-5) dy


[(y^-4)/-4] = [(sin^6θ)/6] + C

You can simply further more if you like.

Differential Calculus – Basic Derivation of Polynomials – Part 1

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  1. bryll says:


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