Ordinary Differential Equations:
Ordinary Differential Equations:
When we lay our hands on calculus,it is not easy to ignore a topic like Ordinary Differential Equation.Let us now learn about the basic meaning of Differential Equation.
Definition:
An equation involving independent variables dependent variable and their derivatives is called a differential equation.
Ordinary differential equation:
A differential equation which involves only one independent variable is called an ordinary differential equation.
Order of a differential Equation:
* is the order of the derivative of the highest order, occurring in the differential equation.
Degree of a differential equation:
* is the degree of the highest order differential coefficient appearing in it, after all the differential coefficients are free from radical powers.
* To form a differential equation from a given equation in x, y and containing arbitrary constants. The given equation is differentiated successively as many times as the number of arbitrary constants. These equations are used to eliminate the arbitrary constants and the equation obtained is the required differential equation.
We usually come across questions like how to find the solution of differential equation,the simplest answer to this is a Solution of a differential equation can be found with the following steps.
* A functional relation between x and y which satisfies the given differential equation.
* Solution of a differential equation by the method of variables separable.
The concept of Differential Equation will be clear when we look at some examples:
1) Solve the differential equation dy / dx – 3xy = x
Solution:
Here P(x) = -3x and Q(x) = x
u(x) = e ∫ P(x) dx
= e ∫ -3x dx
= e – x^3
Substitute the value of u(x) and Q(x) in the equation
= e –x^3 y = ∫x e –x^3 dx
= e –x^3 = - (1/2) e –x^3 + c, where c is the constant of integration
y = c e –x^3 – ½
2) Differential Equation Preparation Example 2:
X3-3x2+14x+7.
Solution:
=3x2-6x+14.
3) Differential Equation Preparation Example 3:
Differentiate the given equation:
6x5+3x3+4.
Solution:
= 30x4+9x2.
Hope you like the above example of Ordinary Differential Equation.Please leave your comments, if you have any doubts.
Ordinary Differential Equations:
When we lay our hands on calculus,it is not easy to ignore a topic like Ordinary Differential Equation.Let us now learn about the basic meaning of Differential Equation.
Definition:
An equation involving independent variables dependent variable and their derivatives is called a differential equation.
Ordinary differential equation:
A differential equation which involves only one independent variable is called an ordinary differential equation.
Order of a differential Equation:
* is the order of the derivative of the highest order, occurring in the differential equation.
Degree of a differential equation:
* is the degree of the highest order differential coefficient appearing in it, after all the differential coefficients are free from radical powers.
* To form a differential equation from a given equation in x, y and containing arbitrary constants. The given equation is differentiated successively as many times as the number of arbitrary constants. These equations are used to eliminate the arbitrary constants and the equation obtained is the required differential equation.
We usually come across questions like how to find the solution of differential equation,the simplest answer to this is a Solution of a differential equation can be found with the following steps.
* A functional relation between x and y which satisfies the given differential equation.
* Solution of a differential equation by the method of variables separable.
The concept of Differential Equation will be clear when we look at some examples:
1) Solve the differential equation dy / dx – 3xy = x
Solution:
Here P(x) = -3x and Q(x) = x
u(x) = e ∫ P(x) dx
= e ∫ -3x dx
= e – x^3
Substitute the value of u(x) and Q(x) in the equation
= e –x^3 y = ∫x e –x^3 dx
= e –x^3 = - (1/2) e –x^3 + c, where c is the constant of integration
y = c e –x^3 – ½
2) Differential Equation Preparation Example 2:
X3-3x2+14x+7.
Solution:
=3x2-6x+14.
3) Differential Equation Preparation Example 3:
Differentiate the given equation:
6x5+3x3+4.
Solution:
= 30x4+9x2.
Hope you like the above example of Ordinary Differential Equation.Please leave your comments, if you have any doubts.
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