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Differential Equation is an equation consisting of the dependent and independent variables. The derivatives of the function are responsible for a change of a function at a particular point. These equations have a variety of applications, in Physics, Biology, Anthropology, Geology, Economics, etc. Differential equations are mainly used to simplify the solutions that satisfy the properties and the equations of the solutions. We can solve differential equations with the help of different formulas. Let us discuss the order, degree, and types of differential equations with real-life examples.
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Key Takeaways: Differential equations, Variables, Dependent and independent variables, Differential formulas, Derivatives, Equation
What is a Differential Equation?
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A differential equation is defined as an equation consisting of derivatives of the dependent variable with respect to the independent variable.
dy/dx= f(x)
Some of the examples of differential equations are:
X2 – 5x + 5 = 0
sin x + cos x = 0
x + y = 5
These equations involve independent or dependent variables only, but the below equation involves variables as well as derivatives of the dependent variable ‘y’ with respect to the independent variable ‘x’.
x*dy/dx + y = 0
Read More: Differentiation and Integration Formula
Order of a Differential Equation
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Order of a Differential Equation is defined as the ‘order of the highest order’ derivatives of the dependent variable with respect to the independent variable in the differential equation.
dy/dx = ex
(highest derivative of first order)
d2y/dx2 + y = 0
(highest derivative of second order)
d3y/dx3 + x2 (d2y/dx2)3 = 0
(highest derivative of third order)
Degree of Differential Equation
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Degree of Differential Equation is defined as the ‘power of the highest order’ where the original equation is in the form of a polynomial equation in derivatives such as y’,y”,y”’ etc.
d3y/dx3 + 2(d2y/dx2) – dy/dx + y = 0
(polynomial equation in y”’,y”,y’)
(dy/dx)2 + (dy/dx) –sin2y = 0
(polynomial equation in y’)
dy/dx + sin(dy/dx) = 0
(not a polynomial equation in y’ and degree of such differential equation cannot be defined)
Types of Differential Equations
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The different types of differential equations are:
- Ordinary Differential Equations: A differential equation that involves derivatives of the dependent variable with respect to an independent variable is called an ordinary differential equation. For example:
2d2y/dx2 + (dy/dx)3 = 0
- Linear Differential Equations: It is an equation having a variable, a derivative of this variable, and a few other functions.
dy/dx + Py = Q, (where P and Q are constant)
dy/dx + Py = Q (linear differential equation in y)
dx/dy + P’x = Q’ (linear differential equation in x)
- Non-Linear Differential Equations: It is an equation that contains nonlinear terms, is called nonlinear differential equations.
d sin(y)/dx + xy dy/dx = sin x
- Homogeneous Differential Equations: If the degree of f(x,y)dy = g(x,y)dx, then the equation is known as a homogeneous differential equation.
F(x,y) = 2x-8y
If we replace x and y with ‘vx’ and ‘vy’ respectively,
F(vx, vy) = 2vx – 8vy = v(2x-8y) = vF(x,y)
Here, the function can be written in the form of vnF(x,y)
- Non Homogeneous Differential Equations: It is an equation wherein the degree of f(x,y)dy and g(x,y) dx is not the same.
F(x,y) = sin x + cos y
If we replace x and y with ‘vx’ and ‘vy’ respectively,
F(vx, vy) = sin(vx) + cos(vy) ≠ vnF(x,y)
- Partial Differential Equations: It is an equation in which the equation contains unknown multivariable with their partial derivatives. It is expressed as:
F(x1, x2,……xm, u, ux1,……uxm) = 0
Read More: Formation of a Differential Equation
Formulas
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- General form of linear differential equation:
- d(x + y) = dx + dy
- d(xy) = y dx + x dy
- d(x/y) = y dx-x dy/ y2
- d(y/x) = x dy-y dx/ x2
- d(ln xy) = ydx + xdy/ xy
- d(ln y/x) = (xdy – ydx)/xy
- d(0.5 ln x+y/ x-y) = xdy – ydx/ x2 – y2
- d(tan-1 y/x) = xdy – ydx/ x2 + y2
- d( x2 + y2)1/2 = xdx + ydy/ (x2 + y2)½
- Extended form of linear differential equation:
- dv/dx + (1-n) v.P = Q(1-n)
Applications of Differential Equation in real life
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Differential Equations have several applications in various fields like engineering, mathematics, science, etc.
- They are used in medical science for modelling cancer growth or the spread of disease in the body.
- To calculate the growth and decay of the population of any country.
- It is used to find out Newton’s law of Cooling, Newton’s 2nd Law of Motion.
- Electricity movement can be determined with the help of a differential equation.
- To calculate glucose absorption by the body.
- The motion of a pendulum or ocean waves can also be described with this equation.
- It is used in engineering subjects like Strength of Materials to find out the calculations of a bridge, beam and pillars, Thermodynamics etc.
Things to Remember
- Ordinary Differential Equations (ODE) have a single independent variable ‘y’.
- Partial Differential Equations (PDE) have two or more independent variables.
- Before solving any differential equation, one must look after the order and degree of the equation.
- Be careful don’t confuse between order and degree as they are the basic criteria to solve any differential equation.
- Degree of Differential Equation is defined as the ‘power of the highest order’ where the original equation is in the form of a polynomial equation in derivatives such as y’,y”,y”’.
- Order of a Differential Equation is defined as the ‘order of the highest order’ derivatives of the dependent variable with respect to the independent variable in the differential equation.
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Sample Questions
Ques. Find out the degree and order of differential equations d3y/dx3 + sin y”’ = 0. (3 marks)
Solution: According to the question, differential equation is,
d3y/dx3 + sin y”’ = 0
y”’+ sin y”’ = 0
The highest order derivative in the differential equation is y’”, so here the order is three. Hence, we can say that the given differential equation is not a polynomial equation in its derivatives and so, its degree is not defined.
Ques. Find out the order and degree of differential equation y’” + 2y” + y’ = 0. (3 marks)
Solution: According to the question differential equation is y’” + 2y” + y’ = 0
The highest order derivative present in the differential equation is y”’, so here the order is three.
Hence, we can say that the given differential equation is a polynomial equation in y”’, y” and y’.
Power of y”’ is 1.
Hence, the degree is one.
Ques. Find out the order and degree of the differential equation d2y/dx2 = cos 3x + sin 3x. (3 marks)
Solution: According to the question given differential equation is d2y/dx2 = cos 3x + sin 3x
d2y/dx2 – cos 3x – sin 3x = 0
The highest order derivative present in the differential equation is d2y/dx2, so here the order is two.
Therefore, we can say that the given differential equation is a polynomial equation in d2y/dx2 and the power is one.
Ques. Verify that the given function is a solution of the corresponding differential equation. (5 marks)
y = x2 + 2x + C : y’ -2x -2 = 0.
Solution: According to the question, it is given that y = x2 + 2x + C
Now differentiating both sides with respect to x, we get
y’ = d/dx (x2 + 2x + C)
y’ = 2x + 2
Then, substituting the values of y’ in the differential equation, we get
= y’ – 2x -2
= 2x + 2 - 2x -2 = 0
LHS = RHS
Therefore, the given function is a solution to the given differential equation.
Ques. Verify that the given function is a solution of the corresponding differential equation. (5 marks)
Y = cos x + C : y’ + sin x = 0
Solution: According to the question, it is given that Y = cos x + C
Now differentiating both sides with respect to x, we will get
Y’ = d/dx (cos x +C)
Y’ = – sin x
By substituting the values of y’ in the given differential equation,
= y’ + sin x
= – sin x + sin x = 0
LHS = RHS
Hence, the given function is a solution to the given differential equation.
Ques. Verify that the given function is the solution of the differential equation. (5 marks)
Y = Ax : xy’ = y (x ≠ 0)
Solution: According to the question, it is given that y = Ax
By differentiating both sides with respect to x we get,
Y’ = d/dx (Ax)
Y’ = A
Now by substituting the values of y’ in the given differential equations,
= xy’
= x * A
= Ax
= Y (given in the question)
LHS = RHS
Hence, the given function is a solution to the given differential equation.
Ques. Find the general solution of the following differential equation dt/dx = (1 + x2) ( 1+ t2) (3 marks)
Solution: The given differential equation is dt/dx = (1 + x2) ( 1+ t2)
dt( 1+ t2) = (1 + x2)dx
By integrating both sides of the above equation, we get
∫dt/( 1+ t2) = ∫(1 + x2)dx
tan-1 t = ∫dx ∫dx2
tan-1 t = x + x3/x + C
This is the required general solution of the differential equation.
Ques. How many arbitrary constants are there in the particular solutions of a differential equation of third-order? (3 marks)
Solution: As we already know that the number of constants in the general solution of a differential equation of order n is equal to its order.
So, the number of constants in the general equation of the fourth-order differential equation is four. The solution obtained from the general solution by giving particular values to the arbitrary constants is known as the particular solution of the differential equation. Therefore, there are 0 arbitrary constants in the particular solutions on a differential equation of third order.
Ques. Find the highest order of the differential equation 7x2 * d2/dx2 – 9 dy/dx + y = 0. (3 marks)
Solution: In the above differential equation 7x2 * d2y/dx2 – 9 dy/dx + y = 0
= 7x2y” – 9y’ + y = 0 (the above equation can be written as)
The highest order derivative of the differential equation is y”, so here the order is two.
Hence, we can say that the given differential equation is a polynomial equation in its derivatives of order 2 and the correct option is c) 2.
Ques. Find out the order of the differential equation 10 * d2y/dx2 + 8dy/dx + 4 = 0. (3 marks)
Solution: In the above differential equation 10 * d2y/dx2 + 8dy/dx + 4 = 0
It can be written as: 10y” + 8y’ + 4 = 0
The highest order derivative of the differential equation is y”, so here the order is two
Hence, there is no value of the coefficient in the differential equation. We can say that the given differential equation is a polynomial equation in its derivatives of order 2 and the correct answer is 2.
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