solve the differential equation
$$\frac{dy}{d\theta}=\frac{e^ysin^2\theta}{ysec\theta}$$
$$\int\frac{y}{e^y}dy=\int\frac{sin^2\theta}{sec\theta}d\theta$$
$$-e^{-y}(y+1)=\frac{sin^3\theta}{3}+C$$
Where from here?

Question

lgbasallote · Answer

isnt that the answer already o.O

anonymous · Answer

Hey Lgba!!!!!
No I have to solve for y

lgbasallote · Answer

you have? well that's a bummer

anonymous · Answer

LOl, that's what I thought the e^y makes it ugly

lgbasallote · Answer

$$\frac{y+1}{e^y} = -\frac{\sin^3 \theta}{3} + C$$
im not doing anything yet..just looking at a diff perspective

anonymous · Answer

Sure

lgbasallote · Answer

well you can ln it..

anonymous · Answer

$$ln|-y(y+1)|$$?

anonymous · Answer

$$yln|y+1|$$

lgbasallote · Answer

wait... this thing is $$\Large \int \frac{\sin^3 \theta}{\sec theta} d\theta$$
right?

anonymous · Answer

no

lgbasallote · Answer

should be $$\frac{\sin^4 \theta}{4} + C$$
then

anonymous · Answer

well according to wolfram it's (sin^3)/3

lgbasallote · Answer

oh wait it's a two

lgbasallote · Answer

nevermind

lgbasallote · Answer

well i officially have no idea lol.

anonymous · Answer

yes you do....let's work on it a little more

anonymous · Answer

would $$\frac1y|y+1|$$ be correct for the left side

lgbasallote · Answer

even when in ln it makes no sense

$$y + \ln (y+1) = \ln (-\frac{\sin^3 \theta}{3} + C)$$

lgbasallote · Answer

$$\ln (e^y (y+1) ) \implies \ln e^y + \ln (y+1)$$

anonymous · Answer

Did I make an error in the initial integration?

lgbasallote · Answer

should be -y

anonymous · Answer

$$\int \frac{y}{e^y}dy$$=?

lgbasallote · Answer

your integral is right

anonymous · Answer

$$y=-1-W \left (\frac{1}{3e} \sin^3 (\theta ) +C\right )$$ Where W(z) is Lambert's W function, the inverse function of z(w)=we^w.

You can do it, but it requires a special function not typically used in lower-level maths. You did your integration right, so you probably just wrote the problem wrong.

Also, it is common practice in diff eqns to leave solutions in their implicit form, so I doubt that your teacher really wanted you to solve for y.

lgbasallote · Answer

exactly :| leave in implicit form lol

anonymous · Answer

Let's start from the beginning
$$\frac{dy}{d\theta}=\frac{e^ysin^2\theta}{ysec\theta}$$

anonymous · Answer

$$-e^{-y}(y+1)=\frac{sin^3\theta}{3}+C$$
I'll leave this as the final answer.