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## Mathematics

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**Session**Differential Equations - 3**Session Objectives**• Linear Differential Equations • Applications of Differential Equations • Differential Equations of Second Order • Class Exercise**The standard form of a linear differential equation of**first order and first degree is where P and Q are the functions of x, or constants. Linear Differential Equations**Linear Differential Equations type-1**where P and Q are the functions of x, or constants.**It is a linear equation of the form**The solution is given by Example – 1**Solve the following differential equation:**It is a linear differential equation of the form Example -2 Solution: The given differential equation is**Solution Cont.**The solution is given by**It is a linear differential equation of the form**Example – 3**Solution Cont.**The solution is given by**Linear Differential Equations type – 2**where P and Q are the functions of y, or constants.**Solve the following differential equation:**It is a linear differential equation of the form Example - 4 Solution: The given differential equation is**Solution Cont.**The solution is given by**Applications of Differential Equations**Differential equations are used to solve problems of science and engineering.**Example - 5**A population grows at the rate of 5% per year. How long does it take for the population to double? Use differential equation for it. Solution: Let the initial population be P0 and let the population after t years be P, then [Integrating both sides]**Hence, the population is doubled in**Solution Cont. At t = 0, P = P0**The slope of the tangent at a point P(x, y) on a curve is**If the curve passes through the point (3, -4), find the equation of the curve. Solution: The slope of the curve at P(x, y) is Example - 6**Solution Cont.**The curve passes through the point (3, –4).**Differential Equations of Second Order**is the required general solution of the given differential equation.**Solve the differential equation:**Example -7 Solution: The given differential equation is**Example –8**Integrating (i), we get**Solution Cont.**Again integrating both sides of (ii), we get