The equation which includes second-order derivative is the second-order differential equation. It is represented as; d/dx(dy/dx) = d2y/dx2 = f”(x) = y”.
Month: June 2022
First Order Differential Equation
You can see in the first example, it is the first-order differential equation that has a degree equal to 1. All the linear equations in the form of derivatives are in the first order. It has only the first derivative such as dy/dx, where x and y are the two variables and is represented as:… Continue reading First Order Differential Equation
Order of Differential Equations
The order of a differential equation is the highest order of the derivative appearing in the equation. Consider the following differential equations, dy/dx = ex, (d4y/dx4) + y = 0, (d3y/dx3) + x2(d2y/dx2) = 0 In above differential equation examples, the highest derivative are of first, fourth and third order respectively.
What are Differential Equations?
A differential equation is an equation that contains at least one derivative of an unknown function, either an ordinary derivative or a partial derivative. Suppose the rate of change of a function y with respect to x is inversely proportional to y, we express it as dy/dx = k/y. In calculus, a differential equation is an equation that… Continue reading What are Differential Equations?
Differential Equations
An equation that contains the derivative of an unknown function is called a differential equation. The rate of change of a function at a point is defined by the derivatives of the function. A differential equation relates these derivatives with the other functions. Differential equations are mainly used in the fields of biology, physics, engineering,… Continue reading Differential Equations
Increasing and Decreasing Functions
By using derivatives, we can find out if a function is an increasing or decreasing function. The increasing function is a function that seems to reach the top of the x-y plane whereas the decreasing function seems like reaching the downside corner of the x-y plane. Let us say we have a function f(x) which… Continue reading Increasing and Decreasing Functions
Maxima, Minima, and Point of Inflection
Application of derivatives is also helpful in finding the maxima, minima, and point of inflection of a curve. Maxima and minima are the peaks and valleys of a curve, whereas the point of inflection is the part of the curve where the curve changes its nature(from convex to concave or vice versa). We can find… Continue reading Maxima, Minima, and Point of Inflection
Tangent and Normal To a Curve
The equation of tangent and normal line to a curve of a function can be calculated by using the derivatives. If we have a curve of a function and we want to find the equation of the tangent to a curve at a given point, then by using the derivative, we can find the slope… Continue reading Tangent and Normal To a Curve
Approximation Value
Derivative of a function can be used to find the linear approximation of a function at a given value. The linear approximation method was given by Newton and he suggested finding the value of the function at the given point and then finding the equation of the tangent line to find the approximately close value… Continue reading Approximation Value
Derivative for Rate of Change of a Quantity
Derivatives are used to find the rate of changes of a quantity with respect to the other quantity. By using the application of derivatives we can find the approximate change in one quantity with respect to the change in the other quantity. Assume we have a function y = f(x), which is defined in the interval [a,… Continue reading Derivative for Rate of Change of a Quantity