Content Strategy Manager
A differential equation is a mathematical equation that connects the derivatives of a function. Functions represent physical quantities in real-world applications, whereas derivatives describe the rate of change of the function about its independent variables. The order and degree of a differential equation are useful in solving the problem. The order and degree of the differential equation may be compared to polynomial expressions, and the order and degree of the differential equation can be used to determine the steps necessary to solve the differential equation and the number of viable solutions.
| Table of Content |
Key Terms: Order, Degree, Linear equation, Non – Linear equation, Quasi Linear, Functions, differential equation
Order and Degree of Differential Equation
[Click Here for Sample Questions]
The type and complexity of a differential equation are determined by the order and degree of the differential equation. A differential equation, like a polynomial equation, has a differential of the dependent variable for the independent variable, and the order and degree of the differential equation are useful in locating the differential equation's solutions.
The order of the differential equation may be determined by first determining the derivatives in the differential equation's provided form. In a differential equation, the various derivatives are as follows.
- First Derivative: dy/dx or y'
- Second Derivative: d2y/dx2, or y''
- Third Derivative: d3y/dx3, or y'''
- nth derivative: dny/dxn, or y''''.....n times
Furthermore, the order of the differential equation is determined by the highest derivative, and the exponent of the greatest derivative determines the differential equation's degree. A differential equation has derivatives of the dependent variable about derivatives of the independent variable, just as a polynomial equation in variable x.
Order of Differential Equation
[Click Here for Sample Questions]
The highest order of the derivative appearing in a differential equation is the order of the differential equation. Consider the differential equations below:
dy/dx = ex, (d4y/dx4) + y = 0, (d3y/dx3)2 + x2(d2y/dx2) + xdy/dx + 3 = 0
The highest derivatives in the instances above are of first, fourth, and third-order, respectively.
First Order Differential Equation
As you can see in the first example, it is a one-degree first-order differential equation. All linear equations in the form of derivatives are in the first order. It only has a first derivative, such as dy/dx, where x and y are the two variables, and is written as dy/dx = f(x, y) = y'.
Second-Order Differential Equation
The second-order differential equation is an equation that includes a second-order derivative. It's written like this: d/dx(dy/dx) = d2y/dx2 = f"(x) = y"
Read More: First Order Differential Equation
Degree of Differential Equation
[Click Here for Sample Questions]
The differential equation's degree is the maximum power of the highest order derivative in a differential equation. A positive integer is usually used as the degree of the differential equation. The differential equation's order must be determined first, followed by its degree. A polynomial expression's variable degree is equivalent to the degree of the differential equation. The following are some examples of differential equations:
- 7(d4y/dx4)3 + 5(d2y/dx2)4+ 9(dy/dx)8 + 11 = 0
- (dy/dx)2 + (dy/dx) - Cos3x = 0
- (d2y/dx2) + x(dy/dx)3 = 0
The degrees of the equations in the above differential equations are three, two, and one, respectively.
Types of Differential equations
[Click Here for Sample Questions]
Differential equations are classified into various categories:
- Ordinary Differential Equations
- Linear Differential Equations
- Nonlinear differential equations
- Partial Differential Equations
- Homogeneous Differential Equations
- Nonhomogeneous Differential Equations
Applications of Differential Equation
[Click Here for Sample Questions]
Some applications of real-time differential equations.
- Differential equations are used to describe how exponential functions grow and decay.
- They can also be used to describe the progression of a return on investment.
- They are used in medical research to stimulate cancer growth and illness spread throughout the body.
- It can also be used to explain how electricity moves.
- They aid economists in selecting the most cost-effective investment alternatives.
Things to Remember
- All of the derivatives in the equation, positive and negative if any, are free of fractional powers.
- The derivatives aren't used in any of the fractions.
- There should be no use of the highest order derivative as a transcendental, trigonometric, or exponential function.
- Any term with the highest order derivative's coefficient should be a function of x, y, or some lower-order derivative.
- It is not always possible to determine the degree of a differential equation.
Also Read:
Sample Questions
Ques. Find the order and degree of the following differential equations. (3 marks)
(a) 4(d3y/dx3) - (d2y/dx2)3 + 5(dy/dx) + 4 = 0
(b) 7(d4y/dx4)2 + 5(d2y/dx2)4+ 9(dy/dx)8 + 11 = 0
Ans. (a) The differential equation has a degree of one and is of order three.
(b) The differential equation has a degree of second and is of order fourth.
Ques. Find the order and degree of the following differential equations. (3 marks)
(a) (y''')2 + x2(y')3 - 2x + 11 = 0
(b) (dy/dx)2 + (dy/dx) - Cos3x = 0
Ans. (a) The differential equation has a degree of second and is of order three.
(b) The differential equation has a degree of second and is of order first.
Ques. Find the order of the given differential equation. (3 marks)
(a) d2y/dx2 + (x3+3x)y = 9
(b) d4y/dx4 + (d2y/dx2)2 – 3dy/dx + y = 9
Ans. (a) The highest derivative's order is 2. As a result, the equation is a second-order differential equation.
(b) The highest order derivative has an exponent of one, and the differential equation is a polynomial equation in derivatives. As a result, this equation's degree is 1.
Ques. Find the Degree of Differential Equation (3 marks)
(a) d2y/dx2 + [ (d2y/dx2)2 ]2 = k2[d3y/dx3]2
(b) d2y/dx2 + cos d2y/dx2 = 5x
Ans. (a) Because the highest derivative is of order 3 and the exponent increased to the highest derivative is 2, the order of this equation is 3 and the degree is 2.
(b) The provided differential equation is not a derivative polynomial equation. As a result, the degree of this equation is unknown.
Ques. Find the Degree of Differential Equation (3 marks)
(a) [d3y/dx3]2 + y = 0
(b) 3(d2y/dx2) + x(dy/dx)3 = 0
Ans. (a) The order of this equation is 3 and the degree is 2.
(b) The differential equation has a degree of one and is of order second.
Ques. Figure out the order and degree of differential equation that can be formed from the equation
√1-x2 + √1-y2 = k(x-y) (4 marks)
Ans. Let x = sin θ and y = sin Φ
So, the given equation can be rewritten as:
\(\sqrt{1 – sin\theta^2} + \sqrt{1 – sin\phi^2} = k(sin \theta – sin \phi)\)
\( \Rightarrow (cos \theta + cos \phi) = k(sin \theta – sin \phi)\)
\(\Rightarrow 2 cos \frac{\theta + \phi}{2} cos\frac{\theta – \phi}{2} = 2 k cos \frac{\theta + \phi}{2} sin \frac{\theta – \phi}{2} \)
\(cot \frac{\theta – \phi}{2} = k\)
\(\theta – \phi = 2cot^{-1} k\)
\(sin^{-1}x – sin^{-1}y = 2cot^{-1} k\)
Differentiating both sides w.r.t. x, we get:
\(\frac{1}{\sqrt{1 – x^2}} – \frac{1}{\sqrt{1 – y^2}}\, \frac{dy}{dx} = 0\)
The differential equation has a degree of 1 and is a first-order differential equation.
Ques. Find the order of this equation sin (dy/dx) d3y/dx3 – log(y) = x2 (3 marks)
Ans. The coefficient of the highest order derivative is just a function of dy/dx, a lower order derivative in this case. As a result, the degree of the equation is defined. Even expanding the trigonometric sine function yields sin x = x–x3/3! + x5/5! – ....., which is a polynomial function with an unlimited number of terms.
Because it's multiplied by d3y/dx3, every term in the expansion will include the term d3y/dx3, keeping the degree of the highest order derivative constant, unlike our previous example. As a result, the differential equation's order is 3 and its degree is 1.
For Latest Updates on Upcoming Board Exams, Click Here: https://t.me/class_10_12_board_updates
Check-Out:
Comments