If dy/dx=xcosx^2 and y=-3 when x=0, when x=pi, y=___.
-3.215
sqrt2
1.647
6
3pi
dy = x cos(x^2) dx
letting u = x^2, we have dy = 1/2 cosu du, so
y = 1/2 sin(x^2) + c
y(0) = -3, so c = -3 and thus
y(x) = 1/2 sin(x^2) - 3
y(π) = 1/2 sin(π^2) - 3 ≈ -2.9
I suspect a typo somewhere ...
To find the value of y when x=π with the given differential equation and initial condition, we need to solve the differential equation and find the equation of the curve in terms of y and x. Then, we can substitute x=π to find the corresponding y value.
Let's start by solving the differential equation:
dy/dx = xcos(x^2)
To do this, we use the separation of variables method:
dy = xcos(x^2) dx
Now we integrate both sides:
∫dy = ∫xcos(x^2) dx
Integrating, we get:
y = ∫xcos(x^2) dx
This integral does not have a simple closed-form solution, so we need to use numerical methods or approximation techniques to evaluate it. However, for the purpose of finding y when x=π, we can use the given initial condition.
Given: y=-3 when x=0
Substituting these values into the equation, we have:
-3 = ∫0cos(0^2) dx
-3 = 0
Since the definite integral equals zero, we can conclude that y=-3 for any value of x.
Therefore, when x=π, y=-3.
So, the correct answer is -3.
None of the options provided ( -3.215, √2, 1.647, 6, 3π ) are correct.