tan−1(4x2y) = x + 5xy2
or
tan (x + 5xy^2) = 4x^2 y
what about it?
To find the solution to the equation tan^(-1)(4x^2y) = x + 5xy^2, we need to isolate the variables x and y.
First, let's rewrite the equation in a more convenient form:
tan^(-1)(4x^2y) = x + 5xy^2
Since tan^(-1) is the inverse of tan, we can take the tangent of both sides to simplify the equation:
tan(tan^(-1)(4x^2y)) = tan(x + 5xy^2)
Next, we use the property tan(tan^(-1)(x)) = x, which allows us to simplify the equation further:
4x^2y = tan(x + 5xy^2)
Now, we can solve for x and y by isolating each variable.
To isolate x:
- Move 5xy^2 to the right-hand side of the equation: 4x^2y - 5xy^2 = tan(x)
- Factor out x: x(4x - 5y^2) = tan(x)
- Divide both sides by (4x - 5y^2): x = tan(x) / (4x - 5y^2)
To isolate y:
- Move x to the right-hand side of the equation: 4x^2y - x = tan(x + 5xy^2)
- Factor out y: y(4x^2 - 1 ) = tan(x + 5xy^2)
- Divide both sides by (4x^2 - 1): y = tan(x + 5xy^2) / (4x^2 - 1)
Now we have expressions for x and y in terms of x, y, and tan(x). However, it is not possible to find a simple, exact solution for x and y in terms of the given equation. The equation is transcendental, meaning it cannot be solved algebraically. Instead, numerical methods such as iteration or approximation techniques can be used to obtain an approximate solution.