If dy/dx=cos^2(πy/4) and y=1 when x=0, then find the value of x when y=3
a) 1/8
b) -π/8
c) -8/π
d) None of these
dy/ cos^2( ay) = dx where a = pi/4
integral dy/cos^2 (ay) = (1/a) tan ay from integral table
so
x = (4/pi) tan pi y/4 + c
0 = (4/pi) tan pi/4 + c
but tan (pi/4) = 1
so
c = -4/pi
and
x = (4/pi) tan pi y/4 - 4/pi = (4/pi) (tan pi y/4 -1)
now put in y = 3
looks like about -8/pi
To find the value of x when y=3, we need to solve the differential equation dy/dx = cos^2(πy/4) for x.
Let's separate variables and integrate both sides of the equation:
∫ 1/cos^2(πy/4) dy = ∫ dx
To integrate 1/cos^2(πy/4), we can use the trigonometric identity cos^2(θ) = (1 + cos(2θ))/2:
∫ 1/cos^2(πy/4) dy = ∫ 2/(1 + cos((2πy)/4)) dy
Now substitute u = (2πy)/4:
∫ 2/(1 + cos(u)) du = ∫ 2/(1 + cos(u)) du
We can use the half-angle formula for cosine to simplify the integral:
∫ 2/(1 + cos(u)) du = 2∫ 2/((1 + cos^2(u)/2) du
= 4∫ du/(1 + cos(u))
To integrate the above expression, we'll use the substitution v = tan(u/2), which gives dv = (1/2)sec^2(u/2) du:
4∫ du/(1 + cos(u)) = 8∫ dv/(1 + (1 - v^2)/(1 + v^2))
= 8∫ dv/((1 + v^2) + (1 - v^2))
= 8∫ dv/2
= 4v
Substituting back u = (2πy)/4:
4v = 4tan((πy)/4)
Now integrating both sides, we get:
4v = 4tan((πy)/4)
∫ dx = ∫ 4tan((πy)/4) dy
Integrating both sides of the equation gives:
x = 4∫ tan((πy)/4) dy + C
To find the constant term C, we can use the initial condition y = 1 when x = 0:
0 = 4∫ tan((π*1)/4) dy + C
0 = 4∫ tan(π/4) dy + C
0 = 4(∫ 1 dy) + C
0 = 4y + C
Since y = 1 when x = 0, we substitute these values into the equation:
0 = 4(1) + C
C = -4
Therefore, the equation becomes:
x = 4y - 4
To find the value of x when y = 3:
x = 4(3) - 4
x = 12 - 4
x = 8
Hence, the value of x when y = 3 is 8.
So, the correct answer is:
d) None of these
To solve this problem, we need to integrate the given differential equation:
dy/dx = cos^2(πy/4)
Let's start by separating variables, which means moving all the y-related terms to one side and all the x-related terms to the other side:
1/cos^2(πy/4) dy = dx
Next, we can integrate both sides of the equation. On the left side, we have the integral of 1/cos^2(πy/4) dy, which we can simplify by using the trigonometric identity:
sec^2(x) = 1 + tan^2(x)
Since cos^2(x) is 1/cos^2(x) (the reciprocal of the identity above), we can rewrite the left side as:
dy = sec^2(πy/4) dy = 1 + tan^2(πy/4) dy
Now, the integral becomes:
∫ dy = ∫ (1 + tan^2(πy/4)) dx
The integral of dy is simply y, and the integral of dx is x. So we have:
y = x + ∫ tan^2(πy/4) dx
To find the value of x when y = 3, we need to evaluate the integral and substitute y = 3:
3 = x + ∫ tan^2(π(3)/4) dx
The next step is to evaluate the integral. We can use a trigonometric identity to simplify it:
tan^2(π(3)/4) = sec^2(π(3)/4) - 1
Now, we can substitute this back into the equation:
3 = x + ∫ (sec^2(π(3)/4) - 1) dx
At this point, we can integrate the equation. The integral of sec^2(x) is tan(x), and the integral of 1 is x. So we have:
3 = x + [tan(π(3)/4) - x]
Simplifying further:
3 - x = tan(3π/4)
Now, we need to find the value of x when y = 3. We can use the initial condition given in the problem: y = 1 when x = 0. Using this initial condition, we can solve for the constant of integration (C):
1 = 0 + [tan(π(1)/4) - 0] + C
1 = tan(π/4) + C
C = 1 - tan(π/4)
Substituting this value of C back into our equation:
3 - x = tan(3π/4) + 1 - tan(π/4)
To find the value of x, we can simplify the equation further:
3 - x = 1 - tan(π/4) + tan(3π/4)
To determine the value of x, we can rearrange the equation:
x = 3 - 1 + tan(π/4) - tan(3π/4)
Now, we just need to calculate this expression:
x = 2 + 1 - 1 = 2
Therefore, the value of x when y = 3 is x = 2.
Since none of the answer choices provided match x = 2, the correct answer would be:
d) None of these