Explanation

To find the center of the circle, rewrite the equation in standard form:

45x² - 60x + 45y² + 36y = -19

Divide by 45:

x² - (4/3)x + y² + (4/5)y = -19/45

Complete the square:

(x - 2/3)² + (y + 2/5)² = ...

The center is (2/3, -2/5).

Explanation

Given the product of two complex numbers (x + iy) and (x' + iy') is zero:

(x + iy)(x' + iy') = 0

This implies either (x + iy) = 0 or (x' + iy') = 0.

Therefore, at least one of the complex numbers must be zero.

Explanation

The Laplace equation is a 2nd order partial differential equation, given by:

∇²u = 0 or ∂²u/∂x² + ∂²u/∂y² + ∂²u/∂z² = 0

Explanation

Explanation

The differential equation xy' = y can be rewritten as y'/y = 1/x.

This equation has a unique solution when x ≠ 0, since dividing by zero is undefined.

The solution exists on the region defined by either x > 0 or x < 0.

Explanation

To find g(3), substitute t = 3 into the equation:

g(3) = (3)³ + (3)² - 9(3) - 9

= 27 + 9 - 27 - 9

= 0

Explanation

Given dt/dx = xy^(1/2)

Separate variables:

dt/y^(1/2) = x dx

Integrate both sides:

∫y^(-1/2) dy = ∫x dx

2y^(1/2) = (x²)/2 + C

y^(1/2) = x²/4 + C'

Square both sides:

y = (x²/4 + C')²

For simplicity, let's assume C' = 0:

y = (x²/4)² = x⁴/16

Explanation

The argument of the product of two complex numbers is the sum of their arguments.

This property comes from the multiplication of complex numbers in polar form.

Explanation

Given ∫12x⁻⁵ √(-3x⁻⁴ + 1) dx

Let u = -3x⁻⁴ + 1

du/dx = 12x⁻⁵

du = 12x⁻⁵ dx

∫√u du = ∫u^(1/2) du

= (2/3)u^(3/2) + C

Substituting back u = -3x⁻⁴ + 1:

= (2/3)(-3x⁻⁴ + 1)^(3/2) + C

Explanation

Given y = 2/x

Let's find the first and second derivatives:

dy/dx = -2/x²

d²y/dx² = 4/x³

Now, let's evaluate y + 1/2 * d²y/dx²:

y + 1/2 * d²y/dx² = 2/x + 1/2 * 4/x³

= 2/x + 2/x³

= (2x² + 2)/x³

= 2(x² + 1)/x³

= 2(1 + x²)/x³