Differential Equations Formula

A differential equation by definition is an equation that contains one or more functions with its derivatives. The rate of change of a function at a point is defined by the derivatives of the function. Differential equations are mainly used in the fields of biology, physics, engineering, and many. The main purpose of the differential equation is for studying the solutions that satisfy the equations and the properties of the solutions. In this article, we will learn how to solve a differential equation. Explicit formulas help us to solve the differential equation. In this lesson, we shall discuss the definition, types, methods to solve the differential equation, order, and degree of the differential equation, types of differential equations, with real-world examples, and practice problems.

What Is the Differential Equations Formula?

An equation that contains the derivative of a function is called a differential function. In calculus, a differential equation is an equation that involves the derivative (derivatives) of the dependent variable with respect to the independent variable (variables). The differential equations formulas help us to find the solution of the derivatives. They include higher-order differentials such as dⁿy/dxⁿ. There are four important formulas for differential equations to find the order, degree of the differential equation, and to work across homogeneous and linear differential equations.

First-order linear differential equations take the form

\[\frac{{dy}}{{dx}} + P(x)y = Q(x)\]

where P and Q are functions of x alone.

Formula 1

The order of a differential equation is the order of the highest order differential equation. Similar to the general equations with a variable x, here we have the differential dy/dx, which is written with varying degrees as exponents. 

dⁿy/dxⁿ + ......d³y/dx³ + d²y/dx²+ dy/dx + k = 0

where n = the order of the differential equation.

Formula 2
The degree of a differential equation is the degree of the highest order derivative.

(dⁿy/dxⁿ)^a + .....(.d³y/dx3)⁴ + (d²y/dx²)²+ (dy/dx) + k = 0

where n = order of the differential equation

a = the degree of the differential equation is 'a'.

Formula 3

By definition, a homogeneous function \(f\left( {x,y} \right)\) of degree n satisfies the property

\[f\left( {\lambda x,\lambda y} \right) = {\lambda ^n}f\left( {x,y} \right)\]

Homogenous Differential Equation refers to an equation in which the substitution of x, y with λx, λy, can be manipulated to get λⁿ common for the entire expression.

f(λx, λy) = λⁿf(x, y) for all λ ∈ R

Formula 4
Linear Differential Equation is similar to a normal equation, but with a variation of the variables. Rather than x and y as variables, here we have dy/dx and y as variables. This is a linear differential equation in y and P and Q are the constants or expressions in 'x'.

dy/dx + Py = Q is the differential equation.

General solution of the differential equation is \(y = e^{-\int P.dx}.\int(Q.e^{\int P.dx}).dx + C\)

\(e^{-\int P.dx\) is the integrating factor.

Formula 5

A partial differential equation (PDE) is a differential equation that contains partial derivatives of the dependent variable(s) with more than one independent variable. 

A first-order partial differential equation with n independent variables has the general form
\(F\left(x_{1}, x_{2}, \ldots, x_{n}, w, \frac{\partial w}{\partial x_{1}}, \frac{\partial w}{\partial x_{2}}, \ldots, \frac{\partial w}{\partial x_{n}}\right)=0\)
where \(w=w\left(x_{1}, x_{2}, \ldots, x_{n}\right)\) is the unknown function and \(F(\ldots)\) is a given function.

The simple PDE is given by;

∂u/∂x (x,y) = 0 

The above relation implies that the function u(x,y) is independent of x which is the reduced form of the partial differential equation formula stated above. The order of PDE is the order of the highest derivative term of the equation.

Let us check out a few solved examples to understand more about differential equation formulas.

Examples Using Differential Equations Formula

Example 1: What is the order of the differential equations (d²y/dx²) + x(dy/dx) + y = 2sinx? 

Solution: The order of the given differential equation (d²y/dx²) + x(dy/dx) + y = 2sinx is 2.

Answer: The order is 2

Example 2: Find the order and degree of the differential equation (d³y/dx³)² + (d²y/dx²)³ + (dy/dx)² + 5 = 0.

Solution:

The given differential equation is (d³y/dx³)² + (d²y/dx²)³ + (dy/dx)² + 5 = 0

Order: It is the order of the highest derivative in the equation, and it is 3

Degree: It is the power of the highest derivative in the equation, and it is 2.

Answer: Hence the order is 3 and the degree is 2.

Example 3: Find the general solution of the differential equation dy/dx = 4/x.

Solution: 

The given differential equation is dy/dx = 4/x
To find the general solution we need to separate the x and y terms and then integrate them on both sides.

dy/dx = 4/x
dy = 4/x.dx
Integrate on both sides.
\(\int \).dy = \(\int \) 4/x.dx
y = 4logx + C

Answer: Therefore the general solution is y = 4 logx + c.