A differential equation is an equation that contains an independent variable, a dependent variable, and the derivatives of the dependent variable. For example, dy/dx = 25x is a differential equation because it contains a dependent variable, ‘y’, and an independent variable, ‘x’, along with the function of x.
The order of the differential equation is defined as the highest-order derivative appearing in a differential equation, which means how many times the differentiation is performed in the equation. dy/dx = f(x) is a differential equation of order 1. y” + 2y’ – 20 = 0 is of order 2. Whereas, the strength of the highest-order derivative appearing in a differential equation after it has been freed of radicals and fractions is referred to as the differential equation's degree. In simpler terms, the power of the larger derivative is known as a degree. There is no defined degree if the differential equation consists of logarithmic, exponential, or trigonometric functions. For example: (y”)2 – y’ = 21 is a differential equation of degree 2.
A solution of a differential equation is of the type k = f(l)+C if it satisfies the equation. There are particularly two types of solution for any differential equation. Assume you're given a differential equation of order n. It is referred to as a general solution if its solution comprises n arbitrary constants. By assigning specific values to arbitrary constants in the general solution of a differential equation, we can get its specific solutions.
We derive the value of C by plugging x equivalent to a and into the general solution. We find the specific solution of the specified differential equation with this value constant as C. The Variable separable method is utilized to find the general solution of any given equation. First, we send all the like times to one side of the equation such that the equation becomes of the form dy/dx. Then dy and dx are integrated separately on both sides. The standard integration formulas and theories are used to find the integration of the dx side. Finally, we are left with an equation summed to a constant value.
The majority of the differential equations in this world are linear differential equations of the form (dy/dx) + Py = Q. Where P is a numeric constant and Q is an algebraic constant or a function of variable x. The opposite of the equation is also possible, dx/dy + Px = Q, where Q is an algebraic constant or a function of variable y. Solving a linear differential equation is quite fascinating. First, we determine the integrating factor of the equation. Integrating factor is the exponential raised to the power of integration of P. This value often comes as logarithmic, so it cancels the exponent part. Then the solution of the equation equals the product of y and the integrating factor. This is often the integration of Q and integrating factors (I.F.). A constant term C is bound to be produced after the solution.