If dy/dx = x (cos x^2) and y = −3 when x = 0, when x = pi, y = .
A. −3.215
B. sqrt(2)
C. 1.647
D. 6
E. 3pi
For some odd reason, I keep getting -0.5325989. I can't come up with any of the answers given.
I=∫ xcos(x²)dx
sub u=x², du=2xdx
I=∫cos(u)du/2
=sin(u)/2+C
When x=0, u=0, y=-3 => C=3
so
y=sin(x²)/2-3
So calculate
y=sin(π²)-3
Note that if you use your calculator, make sure you put the angle to be in radians mode to get the correct answer.
y=sin(pi^2)-3 yields approximately -3.430301217, which still doesn't fit the bill for the options I am given.
Sorry that there was a typo (I forgot to divide sin(x²) by 2):
y=sin(x²)/2-3
So calculate
y=sin(π²)/2-3
To find the value of y when x = π, we need to solve the given differential equation and substitute the known values.
First, let's integrate both sides of the equation dy/dx = x(cos(x^2)) with respect to x:
∫ dy = ∫ x(cos(x^2)) dx
Integrating the left side, we get y + C1 (the constant of integration).
For the right side, we can use the substitution u = x^2, du = 2x dx:
∫ cos(x^2) dx = 1/2 ∫ cos(u) du = 1/2 sin(u) + C2
Substituting back, we get:
y + C1 = 1/2 sin(x^2) + C2
Now, we can use the initial condition y = -3 at x = 0 to find the values of the constants C1 and C2:
-3 + C1 = 1/2 sin(0^2) + C2
-3 + C1 = 1/2(0) + C2
-3 + C1 = C2
We can substitute C2 = -3 + C1 back into the equation:
y + C1 = 1/2 sin(x^2) + (-3 + C1)
y = 1/2 sin(x^2) - 3
Now, we can find the value of y at x = π:
y = 1/2 sin(π^2) - 3
y = 1/2 sin(π^2) - 3
Using a calculator or the trigonometric values, we find that sin(π^2) is approximately 0.1558.
y = 1/2 * 0.1558 - 3
y = 0.0779 - 3
y ≈ -2.9221
From the given answer options, none of them matches the value we obtained (-2.9221), which might explain why you are getting a different result. It's possible that there was an error during calculation or in the initial equation provided.
To double-check, you can review the equation and initial conditions to ensure accuracy.