Find the particular solution to y ′ = sin(x) given the general solution is y = C − cos(x) and the initial condition y(pi/2)=1
well, just plug in the given condition
C-cos(pi/2) = 1
C = 1
So,
y = 1-cos(x)
Well, well, well, it seems like you're dealing with a differential equation. Don't worry, I'll do my best to solve it for you!
First, let's start by finding the derivative of the general solution y = C - cos(x). The derivative of y with respect to x is y' = 0 + sin(x) = sin(x).
Now we need to apply the initial condition y(pi/2) = 1 to find the particular value for the constant C. Plugging in x = pi/2 and y = 1 into the general solution, we get:
1 = C - cos(pi/2)
1 = C - 0
C = 1
So our particular solution is y = 1 - cos(x). And with that, my friend, we're done! Keep in mind that if you have any more questions or just need a good laugh, I'm always here to help.
To find the particular solution to the differential equation y ′ = sin(x), we need to find the constant C using the given initial condition y(pi/2) = 1.
The general solution to the given differential equation is y = C − cos(x).
To find C, substitute the initial condition into the general solution:
1 = C - cos(pi/2)
Simplifying:
1 = C - 0
Thus, C = 1.
Therefore, the particular solution to the differential equation y ′ = sin(x) with the initial condition y(pi/2) = 1 is:
y = 1 − cos(x)
To find the particular solution to the differential equation y ′ = sin(x) with the given general solution y = C − cos(x) and the initial condition y(pi/2)=1, follow these steps:
Step 1: Start by substituting the given values into the general solution.
In this case, the initial condition y(pi/2) = 1 means we can substitute x = pi/2 and y = 1 into the general solution y = C − cos(x).
So, we have 1 = C − cos(pi/2), which simplifies to 1 = C + 1.
Step 2: Solve for the constant C.
To solve for C, subtract 1 from both sides of the equation: C = 1 - 1, which gives C = 0.
Step 3: Substitute the value of C back into the general solution.
Now that we have the value of C, we can substitute it back into the general solution y = C − cos(x) to get the particular solution.
The particular solution is y = 0 − cos(x), which simplifies to y = -cos(x).
So, the particular solution to the given differential equation with the initial condition y(pi/2) = 1 is y = -cos(x).