Solve the differential equation. (Use C for any needed constant. Your response should be in the form 'g(y)=f(θ)'.)

Question

Solve the differential equation. (Use C for any needed constant. Your response should be in the form 'g(y)=f(θ)'.)

7(dy/d(theta))= ((e^ysin^2(theta))/(ysec(theta)))

Answer 1

This is one of those questions that looks really complicated bu is not so bad when you get into it. In face it's a separable differential equation with well-known integrals.

I will use t to stand for theta just to make things more readable.

Given 7 dy/dt = (e^y sin^2 (t) )/ (y sec t)

Separable: Multiply and divide (only) until we get all the y and dy terms on the left, and all the t and dt terms on the right

(y / e^y) dy = 1/7 ((sin^2 t) / sec t) dt (already better)

Now we remember 1/e^y = e^-y, and that sec t = 1 / cos t so 1/sec t = cos t

(e^-y) y dy = (1/7) sin^2 t cos t dt (much better!!)

Now we integrate both sides.

Left Side -- By Parts

Remember to choose u so that du is as simple as possible, anddvb so that it is possible to integrate.

Let u = y and let dv = e^-y dy

du/dy = dy/dy = 1

so du = dy

dv = e^-y dy

Quick sub

Let z = -y

dz/dy = -1

dz = -dy

Integral of e^-y dy = Int of e^z (-dz) = - Int 6^z dz = - e^z = - e^y

By Parts: Int of u dv = uv - Int of v du

Int of y * e^-y dy = y*(-e^-y) - Int of (-e^-y) dy

-y e^-y + (-e^-y)

Left Side Integral = -e^-y(y + 1)

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Right Side: Substitution

We noie derivative of sin t is cos t.

Let w = sint

dw/dt = cost t

So dw = cos t dt

1/7 Int of sin^2 t cos t dt = 1/7 Int of w^2 dw = (1/7)* (w^3) / 3 = (1/21) sin^3 t + C

[We only need one constant of integration and traditionally we put it on the right.]

Put it all together and simplify as far as possible, clean up to standard form

-(e^-y )(y+1) = (1/21) sin^3 t + C

(e^-y )(y+1) = (-1/21) sin^3 t + C

(e^[-y] )(y+1) = (-1/21) sin^3 [theta] + C

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