Nonlinear Equations

To introduce the subject, we consider a few examples of nonlinear equations: $${x^3} + x + 1 = 0$$ is an algebraic equation; there is only one unknown, but it occurs in the third power. There are three solutions, of which two are conjugate complex.$$2x - \tan x = 0$$is a transcendental equation. Again, only one unknown is present, but now in a transcendental function. There are denumerably many solutions.$$\sin x + 3 \cos x = 2$$is a transcendental equation only in an unessential way, since it can be transformed at once into a quadratic equation for eix. While there are infinitely many solutions, they can all be derived from two solutions through addition of multiples of 2π.$${x^3} + {y^2} + 5 = 0$$$$2x + {y^3} + 5y = 0$$is a system of two nonlinear algebraic equations in two unknowns x and y. It can be reduced to one algebraic equation of degree 9 in only one unknown. This latter equation has nine solutions which generate nine pairs of numbers (x _i ,y _i ), i = 1,…, 9, satisfying the given system. (There are fewer if only real x,y are admitted.)