Linear Equation vs Nonlinear Equation
In mathematics, algebraic equations are equations, which are formed using polynomials. When explicitly written the equations will be of the form P(x) = 0, where x is a vector of n unknown variables and P is a polynomial. For example, P(x,y) = 4x5 + xy3 + y + 10 = 0 is an algebraic equation in two variables written explicitly. Also, (x+y)3 = 3x2y – 3zy4 is an algebraic equation, but in implicit form and it will take the form Q(x,y,z) = x3 + y3 + 3xy2 +3zy4 = 0, once written explicitly.
An important characteristic of an algebraic equation is its degree. It is defined to be the highest power of the terms occurring in the equation. If a term consists of two or more variables, the sum of the exponents of each variable will be taken to be the power of the term. Observe that according to this definition P(x,y) = 0 is of degree 5, while Q(x,y,z) = 0 is of degree 5.
Linear equations and nonlinear equations are a two-partition defined on the set of algebraic equations. The degree of the equation is the factor that differentiates them from each other.
What is a linear equation?
A linear equation is an algebraic equation of degree 1. For example, 4x + 5 = 0 is a linear equation of one variable. x + y + 5z = 0 and 4x = 3w + 5y + 7z are linear equations of 3 and 4 variables respectively. In general, a linear equation of n variables will take the form m1x1 + m2x2 +…+ mn-1xn-1 + mnxn = b. Here, xi’s are the unknown variables, mi’s and b are real numbers where each of mi is non-zero.
Such an equation represents a hyper plane in the n-dimensional Euclidean space. In particular, a two variable linear equation represents a straight line in Cartesian plane and a three variable linear equation represents a plane on Euclidean 3-space.
What is a nonlinear equation?
A quadratic equation is an algebraic equation, which is not linear. In other words, a nonlinear equation is an algebraic equation of degree 2 or higher. x2 + 3x + 2 = 0 is a single variable nonlinear equation. x2 + y3+ 3xy= 4 and 8yzx2 + y2 + 2z2 + x + y + z = 4 are examples of nonlinear equations of 3 and 4 variables respectively.
A second degree nonlinear equation is called a quadratic equation. If the degree is 3, then it is called a cubic equation. The degree 4 and degree 5 equations are called quartic and quintic equations respectively. It has been proven that there does not exist an analytic method to solve any nonlinear equation of degree 5, and this is true for any higher degree too. Solvable nonlinear equations represent hyper surfaces that are not hyper planes.
What is the difference between linear equation and nonlinear equation?
• A linear equation is an algebraic equation of degree 1, but a nonlinear equation is an algebraic equation of degree 2 or higher.
• Even though any linear equation is analytically solvable, it is not the case in nonlinear equations.
• In the n-dimensional Euclidean space, the solution space of an n-variable linear equation is a hyper plane, while that of an n-variable nonlinear equation is a hyper surface, which is not a hyper plane. (Quadrics, cubic surfaces and etc.)