Question

What is this type of the equation

− 4 +
4 = 0?
Show by direct substitution that u(x, y) = f(y + 2x) + xg(x +
2x) is a solution for arbitrary functions f and g

Answer 1

This type of equation is a quadratic equation and can be solved using the quadratic formula. To verify if u(x, y) = f(y + 2x) + xg(x + 2x) is a solution for arbitrary functions, we can substitute values into the equation.

Step-by-step explanation:

This type of equation is a quadratic equation of the form at² + bt + c = 0, where a = -4, b = 4, and c = 0. Its solutions can be found using the quadratic formula:

x = (-b ± √(b² - 4ac)) / (2a)

Plugging in the values, we get:

x = (-4 ± √(4² - 4(-4)(0))) / (2(-4))

x = (-4 ± √(16)) / (-8)

x = (-4 ± 4) / (-8)

Simplifying further, we have:

x = (-8 / -8) or x = (0 / -8)

Therefore, the solutions are:

x = 1 or x = 0

To show that u(x, y) = f(y + 2x) + xg(x + 2x) is a solution for arbitrary functions f and g, we can substitute the values of x and y into the equation and check if it holds true.