Calculus: Early Transcendentals, Chapter 9, 9.3, Section 9.3, Problem 8

This equation can be solved by the separation of variables. This means, we can rewrite this equation in a way that all y-containing terms are on the left side and all theta-containing terms are on the right side.
To do this, we can multiply both sides of equation by y and d(theta) :
ydy = e^y * (sin^2(theta))/sec(theta) d(theta)
We can also divide both sides of the equation by e^y :
(ydy)/e^y = (sin^2(theta))/sec(theta) d(theta)
Next, we can simplify both sides of the equation using the rule of exponents:
1/e^y = e^(-y)
and the reciprocal identity: 1/sec(theta) = cos(theta) .
ye^(-y)dy = sin^2(theta)cos(theta)d(theta)
Now we can take the integral of the both sides of the equations.
int (ye^(-y)dy) = int (sin^2(theta)cos(theta)d(theta))
On the left side, we have a product of an exponential function and a polynomial function (y), so this integral will have to be taken by parts.
Let u = y and dv = e^(-y)dy
Then, du = dy and v = int e^(-y)dy = -e^(-y)
According to the integration by parts procedure,
int udv = uv - int (vdu)
Thus, int (ye^(-y)dy = -ye^(-y) - int (-e^(-y))dy = -ye^(-y) - e^(-y)
This will be the left side of the equation after the integration.
On the right side, we have the trigonometric integral which can be solved by substitution. Let u = sin(theta) . Then, du = cos(theta)d(theta) .
Plugging this into the integral, we get
int sin^2(theta)cos(theta)d(theta) = int u^2 * du
This is a power function, integration of which results in u^3/3 + C , where C is an arbitrary constant.
Substituting the expression for u in terms of theta back, we see that the left side after the integration becomes
1/3 sin^3(theta) + C
So, the equation is now
-ye^(-y) - e^(-y) = 1/4 sin^3(theta) + C
We cannot express y(theta) explicitely from here, but we can solve for theta(y) . First, isolate sin^4(theta) :
sin^3(theta) = 3(-ye^(-y) - e^(-y) - C)
Than, take 4th degree root:
sin(theta) = root(4) (-4(ye^(-y) + e^(-y) + C))
Finally, take the inverse sine of both sides:
theta = arcsin(root(3) (-3(ye^(-y) + e^(-y) + C)))
This is the solution of the given equation. Note that it will be defined, depending on the value of constant C, only for restricted values of y.