Consider the differential equation dy/dx=xy-y. Find d^2y/dx^2 in terms of x and y. Describe the region in the xy plane in which all solutions curves to the differential equation are concave down.
just take d/dx(xy-y)
y + xy' - y'
y+(x-1)y'
y+(x-1)(xy-y)
y+y(x-1)^2
y(1+(x-1)^2)
y" y(x^2-2x+3)
to find y,
dy/dx = xy-y = y(x-1)
dy/y = (x-1)dx
ln y = 1/2 (x-1)^2 + c
y = ce^(1/2 (x-1)^2)
= ce^(1/2x^2 - x + 1/2)
= ce^(1/2 x(x-2))
c changes with rearrangement
y is concave down when y" < 0
since e^(f(x)) i always positive, I don't see anywhere where y is concave down, unlesss c<0
To find d^2y/dx^2 in terms of x and y, we need to find the second derivative of y with respect to x.
Given the differential equation dy/dx = xy - y, we can rewrite it as:
dy/dx = x(y - 1)
To find d^2y/dx^2, we need to differentiate both sides of the equation with respect to x. Applying the product rule, we have:
(d/dx)(dy/dx) = (d/dx)(x(y - 1))
Now, let's differentiate the left side and apply the product rule to the right side:
(d^2y/dx^2) = (d/dx)(x(y - 1)) + x(d/dx)(y - 1)
Differentiating x(y - 1) with respect to x:
(d^2y/dx^2) = (d/dx)(xy - x)
= y - 1 + x(dy/dx)
Substituting dy/dx = xy - y, we have:
(d^2y/dx^2) = y - 1 + x(xy - y)
= y - 1 + x^2y - xy
Simplifying further:
(d^2y/dx^2) = x^2y - xy + y - 1
So, the second derivative of y with respect to x is given by d^2y/dx^2 = x^2y - xy + y - 1.
Now, let's determine the region in the xy plane in which all solution curves to the differential equation are concave down. For a curve to be concave down, its second derivative (d^2y/dx^2) must be negative.
In order for x^2y - xy + y - 1 to be negative, the coefficients of y and x^2 need to be negative (assuming y > 0).
Hence, the region in the xy plane in which all solution curves to the differential equation are concave down is given by:
x^2 < 0 and y < 0