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Order and Degree of Differential Equations

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Last updated date: 27th Mar 2024
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How to find the order and degree of differential equations?

A differential equation refers to the mathematical equation that relates some function with its corresponding derivatives. The functions represent some physical quantities in real-life applications, while the derivatives represent the rate of change of the function concerning its independent variables.

 

In two variables, the most general differential equation is of the form - 

f(p, q, q’, q”......) = c, 

where 

  • f(p, q, q’, q”...) is a function of p, q, q’, q”... and so on 

  • p is the independent variable

  • q is the dependent variable 

  • q’, q” are the first-order and the second-order derivative of q respectively

  • c is some constant

 

How to Find the Order and Degree of a Differential Equation? 

The order of a differential equation refers to the highest order derivative involved in that particular differential equation. The degree of a differential equation refers to the exponent or the power of the highest order derivative involved in that particular differential equation, provided that the differential equation satisfies the conditions specified below: 

  • The derivatives in the equation must be free from both negative and positive fractional powers if any 

  • There must be no involvement of the derivatives in any fraction 

  • There must not be any involvement of the highest order derivative as a transcendental, exponential, or trigonometric function. The coefficient of any term in the differential equation containing the highest order derivative should only be a function of p, q, or some lower-order derivative. 

  • If one or more of the conditions mentioned above are not satisfied by the differential equation, then it first needs to be reduced to the form in which it satisfies all of the conditions. In case the equation isn't reducible, then that means it either has no degree or has an undefined degree.

 

Let’s Consider a Few Examples: 

X d2y/dx2 +Y dy/dx + 4y2    

The given differential equation is already in the reduced form. The highest order derivative in this equation is of order 2, and its power or exponent is 1. Therefore, the order of the differential equation is 2 and its degree is 1. 

3y2(dy/dx)3 - d2y/dx2=sin(x/2) 

The highest order derivative involved in this particular differential equation, which is already in the reduced form, is of order 2 and its corresponding power is 1. Therefore, the order of the differential equation is 2 and its degree is 1. 


First Order Differential Equation 

A first order differential equation is linear, when there is only dy/dx and not d2y/dx2, d3y/dx3 and so on, and can be made to look like: 

dy/dx + P(x)y = Q(x), where P(x) and Q(x) are functions of x. 

There’s a special method for solving this equation. 

We use two new functions of x, let them be u and v, and say y=uv. Then, we will solve the equation to find u and v. 

Also, we will find the derivative of y=uv, using the product rule. 

The derivative, dy/dx = udv/dx + vdu/dx (differentiating with respect to x) 


Steps 

1. Substitute y = uv

2. Factor the parts that involve v 

3. Equate the v term with zero and solve using separation of variables to find u 

4. Substitute u into the equation we got at step 2 

5. Solve the equation to find v

6. Substitute u and v into the equation y=uv to find the final answer 


Let’s Understand this Using an Example: 

dy/dx – y/x = 1 

The given differential equation is linear, so let’s follow the steps mentioned above: 

  • Step 1

Substitute y = uv, and dy/dx = u dv/dx + v du/dx 

So, the equation becomes – udv/dx + vdu/dx – uv/x = 1

  • Step 2

Factor the parts involving v 

udv/dx + v( du/dx – u/x ) = 1

  • Step 3 

Equate the v term with 0 

du/dx – u/x = 0 

so, du/dx = u/x

  • Step 4

To find u, solve using separation of variables 

Separate variables: du/u = dx/x 

Put integral sign: ∫ du/u = ∫ dx/x 

Integrate: ln(u) = ln(x) + C 

Make C = ln(k): ln(u) = ln(x) + ln(k) 

So, u = kx

  • Step 5 

Substitute u back into the equation we got at step 2 

kx dv/dx = 1

  • Step 6 

Solve the equation to find v 

Separate variables: k dv = dx/x 

Put integral sign: ∫ k dv = ∫ dx/x 

Integrate: kv = ln(x) + C 

Make C = ln(c): kv = ln(x) + ln(c) 

So, kv = ln(cx) 

And, v = 1/k ln(cx)

  • Step 7 

Now, substitute the values in y = uv, to find the final answers for the original equation 

y = kx 1/k ln(cx) and simplify 

So, the answer is y = x ln(cx) 


Second-Order Differential Equations – Homogeneous With Constant Coefficients 

Let us consider a differential equation of the type y′′+py′+qy=0, where p,q are some constant coefficients.

For each of the equations, we can write the characteristic or auxiliary equation, which is of the form: 

k2+pk+q=0.

The general solution of the homogeneous differential equation depends on the roots of the characteristic quadratic equation. There are 3 different cases, which are as follows: 

1. Discriminant or D of the characteristic quadratic equation is greater than 0, i.e., D>0. Then, the roots of the characteristic equations k1 and k2 are real and distinct. In this case, the general solution is given by the following function: 

y(x)= C1ek1x + C2ek2x 

where C1 and C2 are arbitrary real numbers.


2. Discriminant or D of the characteristic quadratic equation is equal to 0, i.e., D=0. Then, the roots are real and equal. It is said in this case that there exists one repeated root k1 of order 2. The general solution of the differential equation has the following form: 

y(x)=(C1x+C2)ek1x 


3. Discriminant or D of the characteristic quadratic equation is less than 0, i.e., D<0. Such an equation has complex roots k1=α+βi, k2=α−βi. The general solution is written as: 

y(x)=eαx × C1cos(βx)+C2sin(βx).

FAQs on Order and Degree of Differential Equations

1. What are the various types of differential equations?

Differential equations can be divided into the following types:

  • Ordinary Differential Equations

  • Partial Differential Equations

  • Homogeneous Differential Equations

  • Nonhomogeneous Differential Equations

  • Linear Differential Equations

  • Nonlinear differential equations

2. What are the applications of differential equations?

Differential equations are mainly used to compute the function over its entire domain. They can be used to predict the world around us. And hence, they have many applications in fields like applied mathematics, science, and engineering. They have not only technical applications but are used to solve many real-life problems. Here are some of the applications.

  • Differential equations describe many exponential growths and decays.

  • Economists use differential equations to find optimum investment strategies.

  • They are used to describe the change in return on investment over time.

  • In medical science, they are used for predicting the growth of cancer and its spread in the body.

  • In physics, these equations can be used to describe the motion of waves or a pendulum.

  • It is used to describe the movement of electricity.

  • In engineering differential equations are used in­ heat conduction analysis.

  • The ordinary differential equation can be used to establish the relationship between various parts of a bridge.

3. What are ‘integrating factor technique’ and ‘separation of variables’?

There are two methods to find the solution of the differential equation.

Separation of variables

Separation of variables is used when the differential equation can be written in the following form: dy/dx = f(y)g(x).


Here f and g are functions of only y and g respectively. 


Integrating factor

Integrating factor technique is a method used when a differential equation is of this form: dy/dx + p(x)y = q(x) 


Here p and q are both the functions of x.


In this case, the first-order differential equation is of this form: y’+ P(x)y = Q(x). Here P and Q are both functions of x, and also the first derivative of y. 


The higher-order differential equation contains partial or ordinary derivatives of unknown functions and can be represented in any order.

4. What is the order of the differential Equation?

The order of the equation is the order of the highest derivative (also called the differential coefficient) present in the differential equation. The order of the differential equation is 1, then it is called the first order. If the order of the equation is 2, then it is called a second-order, and so on.


For example in the equation below, the order of the highest derivative is 3. So, this is a third-order differential equation.


Example:

\[\frac{d^{3}x}{dx^{3}} + 3x \frac{dy}{dx}  = e^{y}\]


The equation below gives us the second-order differential equation.


\[-(\frac{d^{2}y}{dx^{2}})^{4} + \frac{dy}{dx} = 3\]


In this way, we can get the higher-order differential equations.

5. What is the difference between linear and non-linear differential equations?

A differential equation that has only linear terms of variables (whether unknown or variable) and its derivatives, is a linear differential equation. When the variables and their derivatives are only multiplied by constants, the differential equation is called linear. So, linear just means that the variable in an equation appears only with a power of one. So x is linear, and x3 is non-linear. Any function like cos(x) is also non-linear. 


Linear equations cannot have nonlinear functions, such as trigonometric functions, logarithmic functions, exponential functions, etc. Hence these mentioned functions are a part of nonlinear equations. Another difference is that solutions of linear differential equations create vector space, and the differential operator is also a linear operator in vector space.


Solutions of linear differential equations are relatively easier and can have general solutions. In nonlinear equations, the general solution mostly does not exist and the solution tends to be more problem-specific. Non-linear differential equations are thus a lot more difficult to solve than linear equations.