if y= sin^2(pix), show that the second derivative +4pi^2y=2pi^2

Answer 1

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.