Differential Equations-A Comprehensive Guide for Class 12
I am Suman Mathews, math teacher and content developer. Having taught Mathematics for over three decades, to High school and college students, I can help you with the study of Differential equations. Spend 15 minutes at least reading this and utilise the youtube links at the end of this post which has a variety of free lectures and multiple choice questions.
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First, you need to understand the concept of dependent and independent variable. Given dy/dx, y is the dependent variable and x is the independent variable. Again, in dx/dy, x is the dependent variable and y is the independent variable.
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An equation which involves unknown functions and their derivatives with respect to one or more independent variables is called a differential equation. An equation which involves unknown functions containing only one independent variable and their derivatives with respect to that independent variable is called an ordinary differential equation. In Class 12, we deal with the study of ordinary differential equations.
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The order of a differential equation is the highest order derivative of the dependent variable with respect to the independent variable. If each term involving the derivatives of a differential equation is a polynomial or can be expressed as a polynomial, then the highest exponent of the highest order derivative is called the degree of the differential equation. You need to understand the difference between the general solution and a particular solution of a differential equation.
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The general solution of a differential equation of order one contains one arbitrary constant. Any solution which is obtained from the general solution by giving particular values to the constant is called a particular solution. While solving a differential equation, there are a few types that you need to keep in mind.
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Variables Separables method- In differential equations of this type, you can separate the variables and then integrate. This is the easiest method of integrating a differential equation and can be used in most problems. You can also reduce an equation to the variables separables form by a simple substitution.
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Homogeneous Differential Equation
A differential equation is said to be homogeneous of degree n if if f(lambda x, lambda y)=lambda^n f(x,y)
A differential equation dy/dx=f(x,y) or dx/dy=g(x,y) is said to ne homogeneous if f(x,y) and g(x,y) are homogeneous of degree zero.
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To solve a homogeneous differential equation, we either substitute y = vx or x = vy
We then get dy/dx= v +x dv/dx or dx/dy = x +vdy/dy
This then gets reduced to a variables separables form and we can solve it.
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Linear Differential Equations
A linear equation of the form dy/dx+P(x) y = Q(x) is a linear differential equation.To solve a linear differential equation, you will first need to calculate the Integrating Factor IF = e^int(P(x)dx
Then the general solution is ye^int(P(x)dx=int(q(x)e^int(P(x)dx) +c
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Given the linear differential equation dx/dy+P(y)x=Q(y),
The solution would be xe^int(P(y)dy)=int(Q(y)e^int(P(y)dy)+c
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Need more help?
You can contact me for online classes if you need more help. I can help prepare you for your exams with adequate training in multiple choice questions, concept based questions and Case study based questions.
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Practice tests on Differential equations is also available. There are a number of free resources here in the form of youtube videos. You can access these at no extra cost.