Examples for the fourth order differential equation are,

**Figure-1: Graph of a fourth-order differential equation.**

The degree of the differential equation is the power of the highest-order derivative. Here the original equation is expressed as derivatives such as y’, y”, y”‘, and so on.

## How Do You Solve a Fourth-Order Differential Equation

The steps of solving a fourth-order differential equation are illustrated below

**Step-1:** Write the characteristics equation.

For the equation: y^{4 }+y^{3}+y^{2 }=0 … (Eqn. 1)

The characteristics equation will be m^{4 }+m^{3}+m^{2 }=0 … (Eqn. 2)

**Step-2:** Derive what the characteristics equation is the factor of.

For equation 2, m^{2} (m^{2 }+m^{1}+1)=0 … (Eqn.3)

**Step-3:** Use the **quadratic formula** to derive the value of m.

For equation 3, m=0, m=-1/2±ϳ3

**Step-4:** Finish the solution by solving it for y.

## What Are the Four Types of Fourth-Order Differential Equations

The four types of fourth-order differential equations are:

**Ordinary Differential Equations (ODE)**

In mathematics, an ordinary differential equation (often abbreviated as ODE) is an equation that consists of one or more functions of one independent variable and its derivatives. A differential equation is a mathematical expression that contains a function with one or more derivatives.

#### Example

y^{4}=x^{4}+1

**Partial Differential Equation (PDE)**

A partial differential equation is a mathematical expression that contains an unknown function of two or more variables and its partial derivatives concerning these variables. The highest-order derivatives determine the order of a partial differential equation.

#### Example

∂^{4}u/∂t^{4}=c^{4* }(∂^{4}u/∂x^{4 })

**Linear Differential Equation**

The linear polynomial equation, which consists of derivatives of numerous variables, defines a linear differential equation. When the function is dependent on variables and the derivatives are partial, it is also referred to as Linear Partial Differential Equation.

#### Example

(dy/dx)^{4} + Py^{4} = Q

**Non-linear Differential Equation**

A nonlinear differential equation is not linear in the unknown function and its derivatives. It provides several solutions that might be seen for chaos.

#### Example

[y^{4}(x)]2+y^{3}(x)=f(x^{3})

### What Are the Applications of Fourth-Order Differential Equations?

In real life, fourth-order differential equations are used to compute the movement or flow of electricity, the motion of an object to and fro like a pendulum, and to illustrate thermodynamic ideas. They are also used in medicine to monitor the progression of diseases in graphical form.

## Conclusion

Fourth-order nonlinear differential equations occur naturally in models of physical, biological, and chemical phenomena. Such as elasticity, structural deformation, and soil settlement.

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