if y= sin^2(pix), show that the second derivative +4pi^2y=2pi^2
To find the second derivative of y = sin^2(pi*x), we need to first find the first derivative, and then differentiate that result once more.
First derivative:
y = sin^2(pi*x)
dy/dx = d/dx (sin^2(pi*x))
We use the chain rule, which states that the derivative of a composite function is the derivative of the outer function times the derivative of the inner function:
dy/dx = 2*sin(pi*x) * d/dx (sin(pi*x))
Now we need to find the derivative of sin(pi*x):
d/dx (sin(pi*x)) = cos(pi*x) * d/dx (pi*x)
The derivative of pi*x is simply pi, so:
d/dx (sin(pi*x)) = cos(pi*x) * pi
Now, substitute this back into our expression for dy/dx:
dy/dx = 2*sin(pi*x) * (pi*cos(pi*x))
Now we need to find the second derivative:
d²y/dx² = d/dx (2*sin(pi*x) * (pi*cos(pi*x)))
We will use the product rule, which states that the derivative of a product is the derivative of the first function times the second function, plus the first function times the derivative of the second function:
d²y/dx² = 2 * d/dx(sin(pi*x)) * (pi*cos(pi*x)) + 2*sin(pi*x) * d/dx(pi*cos(pi*x))
We already found the derivative of sin(pi*x) earlier; it is pi*cos(pi*x). Now, we need to find the derivate of pi*cos(pi*x):
d/dx (pi*cos(pi*x)) = -pi^2*sin(pi*x)
Now substitute back into our expression for the second derivative:
d²y/dx² = 2 * (pi*cos(pi*x)) * (pi*cos(pi*x)) + 2*sin(pi*x) * (-pi^2*sin(pi*x))
d²y/dx² = 2*pi^2*cos^2(pi*x) - 2*pi^2*sin^2(pi*x)
Now, we are given that y = sin^2(pi*x), so we can replace sin^2(pi*x) with y in the expression for the second derivative:
d²y/dx² = 2*pi^2*cos^2(pi*x) - 2*pi^2*y
Now we need to show that d²y/dx² + 4*pi^2*y = 2*pi^2:
d²y/dx² + 4*pi^2*y = (2*pi^2*cos^2(pi*x) - 2*pi^2*y) + 4*pi^2*y
Combing the terms:
= 2*pi^2*cos^2(pi*x) + 2*pi^2*y
We can use the trigonometric identity sin^2(x) + cos^2(x) = 1. In this case, our x value is pi*x:
sin^2(pi*x) + cos^2(pi*x) = 1
y + cos^2(pi*x) = 1
Now we can solve for cos^2(pi*x):
cos^2(pi*x) = 1 - y
Substitute this back into our expression for the second derivative:
d²y/dx² + 4*pi^2*y = 2*pi^2*(1 - y) + 2*pi^2*y
= 2*pi^2 - 2*pi^2*y + 2*pi^2*y
= 2*pi^2
Thus, we have shown that the second derivative + 4*pi^2*y = 2*pi^2.