If dy/dx= sin^2(πy/4) and y=1 when x = 0, then find the value of x when y = 3.
A) 0
B) 8/π
C) - π/8
D) None of these
y' = sin^2(πy/4)
dy/sin^2(πy/4) = dx
csc^2(πy/4) dy = dx
Think you can take it from here?
You know what ∫(csc^2 u) du is, right?
In case you get stuck there, the rest is
4/π (-cot π/4 y) = x+c
y = -4/π cot^-1(π/4 (x+c))
Now, y(0) = 1, so
-4/π cot^-1(π/4 c) = 1
cot^-1(π/4 c) = -π/4
since cot^-1(-1) = -π/4,
π/4 c = -1
c = -4/π
y = -4/π cot^-1(π/4 (x-4/π))
y = -4/π cot^-1(π/4 x - 1)
So, when y=3,
-4/π cot^-1(π/4 x - 1) = 3
cot^-1(π/4 x - 1) = -3π/4
cot -3π/4 = 1
so,
π/4 x - 1 = 1
π/4 x = 2
x = 8/π
Thank you very much.
To solve this problem, we need to use calculus techniques to find the value of x when y = 3. Let's start by separating variables and integrating the given differential equation.
Step 1: Separating variables
dy/dx = sin^2(πy/4)
To separate variables, we divide both sides by sin^2(πy/4):
dx/dy = 1/sin^2(πy/4)
Step 2: Integrating both sides
To integrate both sides, we need to find the antiderivative of each side of the equation.
∫dx = ∫1/sin^2(πy/4) dy
The integral of dx is simply x. To evaluate the integral on the right side, we can use a trigonometric identity.
Step 3: Applying a trigonometric identity
Using the identity: 1/sin^2(x) = csc^2(x), we can rewrite the integral as:
x = ∫csc^2(πy/4) dy
Step 4: Finding the antiderivative
The integral of csc^2(u) du is -cot(u) + C, where C is the constant of integration. So, we have:
x = -cot(πy/4) + C
Step 5: Solving for the constant of integration
To find the value of C, we can use the initial condition y = 1 when x = 0.
When x = 0, y = 1, we substitute these values into the equation:
1 = -cot(π/4) + C
The cot(π/4) is equal to 1. Therefore:
1 = -1 + C
C = 2
Step 6: Plug in the value of y and solve for x
Now that we have the value of the constant of integration, C, we can substitute y = 3 into the equation:
x = -cot(π(3)/4) + 2
Before we proceed further, we need to determine the value of cot(3π/4). Using the fact that cot(x) = 1/tan(x):
cot(3π/4) = 1/tan(3π/4)
To evaluate tan(3π/4), we note that the tangent function is negative in the second quadrant, and its absolute value is equal to 1.
tan(3π/4) = -1
Therefore:
cot(3π/4) = 1/(-1) = -1
Plugging this back into the equation:
x = -(-1) + 2
x = 1 + 2
x = 3
So, when y = 3, the value of x is 3. However, this answer is not given as an option.
Therefore, the correct answer is D) None of these.