Using the separation of variables technique, solve the following differential equation with initial condition:

(4x sqrt)(1 - t^2)(dx/dt) - 1 = 0 and x(0)=-2
I believe that the answer is one of the following two options:
a.) 2x^2 = arcsint + 8
b.) arccost + 8 - (1/2)pi

4 x dx = dt/sqrt (1-t^2)

2 x^2 = sin^-1 (t) + c
when t = 0, x = 2
8 = 0 + c
c = 8
so 2 x2 = sin^-1 (t) + 8
or
2 x^2 = cos^-1(t) + c
8 = pi/2 + c
yes, agree

I'm only permitted to choose one option. And seeing as you came up with arcsin first, would this be a better answer over the option with arccos?

They are the same.

cos (pi/2 - x) = sin x

Well, I hate to burst your bubble, but the correct answer is none of the above options. Let's solve the differential equation step by step and see what we get.

First, we rearrange the equation to isolate dx/dt:

(4x sqrt(1 - t^2))(dx/dt) = 1

Now, we can separate variables by multiplying both sides by dt and dividing both sides by 4x sqrt(1 - t^2):

(dx/4x sqrt(1 - t^2)) = dt

Next, we integrate both sides with respect to their respective variables. On the left side, we use substitution by letting u = (1 - t^2):

∫(dx/4x sqrt(u)) = ∫dt

The integral on the left side can be solved using a trigonometric substitution. We let u = sin^2(θ), which gives du = 2sin(θ)cos(θ)dθ and sqrt(u) = sin(θ). The limits of integration change accordingly:

∫(dx/4x sqrt(u)) = ∫(dx/4x sin(θ)) = (1/4)∫(dx/x sin(θ)) = (1/4)∫(dθ/2cos(θ)) = (1/8)∫(dθ/cos(θ))

Using the identity sec^2(θ) = 1 + tan^2(θ), we can rewrite the integral:

(1/8)∫(dθ/cos(θ)) = (1/8)∫(dθ/sec(θ)) = (1/8)∫(sec(θ)/sec^2(θ))dθ = (1/8)∫(sec(θ)/[1 + tan^2(θ)])dθ

Now, performing a substitution by letting u = tan(θ), we have du = sec^2(θ)dθ:

(1/8)∫(sec(θ)/[1 + tan^2(θ)])dθ = (1/8)∫(du/[1 + u^2]) = (1/8)arctan(u) + C

Substituting back for u = tan(θ):

(1/8)arctan(u) + C = (1/8)arctan(tan(θ)) + C = (1/8)arctan(t) + C

Now, we continue solving for x:

(1/8)arctan(t) + C = (1/8)arctan(t) + ln|x| + D

Combining the constants C and D into a single constant:

(1/8)arctan(t) + ln|x| + D = ln|x| + (1/8)arctan(t) + E

Finally, using the initial condition x(0) = -2, we can plug this into the equation:

ln|-2| + (1/8)arctan(0) + E = ln(2) + E = -2

Solving for E:

ln(2) + E = -2
E = -2 - ln(2)

So the final solution is:

ln|x| + (1/8)arctan(t) = ln(2) - 2

Now isn't that more exciting than multiple choice options?

To solve the given differential equation using the separation of variables technique, we need to separate the variables and integrate both sides.

The given differential equation is:
(4x√(1 - t^2))(dx/dt) - 1 = 0

First, let's rearrange the terms:
(4x√(1 - t^2))(dx/dt) = 1

Now, let's separate the variables by putting all x terms on one side and all t terms on the other side:
(1/4x) dx = (1/√(1 - t^2)) dt

Next, integrate both sides with respect to their respective variables:
∫(1/4x) dx = ∫(1/√(1 - t^2)) dt

The integration of the left side can be simplified as:
(1/4) ∫(1/x) dx = (1/4) ln|x| + C1

To integrate the right side, we can use the substitution method by letting u = 1 - t^2. Therefore, du = -2t dt.
The equation becomes:
∫(1/√(1 - t^2)) dt = -1/2 ∫(1/√u) du = -1/2 ∫u^(-1/2) du = -u^(1/2) + C2

Substituting back u = 1 - t^2, we have:
-(1 - t^2)^(1/2) + C2

Now, we can rewrite the equation using the initial condition x(0) = -2:
(1/4) ln|-2| + C1 = -(-1)^(1/2) + C2

Since we have two arbitrary constants C1 and C2, we can combine them into one constant, C:
(1/4) ln|-2| = -(-1)^(1/2) + C

ln(2)/4 = √(-1) + C

Now we can solve for C:
C = ln(2)/4 - i

Therefore, the final solution to the differential equation with the given initial condition x(0) = -2 is:
(1/4) ln|x| = -(1 - t^2)^(1/2) + ln(2)/4 - i

Now, let's check which of the two options a.) or b.) matches the solution we obtained.