Ordinary Differential Equations
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1 Chapter 10 Ordinary Differential Equations 10.1 Introduction Relationship between rate of change of variables rather than variables themselves gives rise to differential equations. Mathematical formulation of most of the physical and engineering problems leads to differential equations. It is very important for engineers and scientists to know inception and solving of differential equations. These are of two types: 1) Ordinary Differential Equations (ODE) 2) Partial Differential Equations (PDE) An ordinary differential equation (ODE) involves the derivatives of a dependent variable w.r.t. a single independent variable whereas a partial differential equation (PDE) contains the derivatives of a dependent variable w.r.t. two or more independent variables. In this chapter we will confine our studies to ordinary differential equations. Prelims: If and are functions of and vanishes after a finite number of differentiations Here is derivative of and is integral of For example = Page 1
2 Order and Degree of Ordinary Differential Equations (ODE) A general ODE of n th order can be represented in the form =0 Order of an ordinary differential equation is that of the highest derivative occurring in it and the degree is the power of highest derivative after it has been freed from all radical signs. The differential equation is having order 3 and degree 1. Whereas is having order 3 and degree 3. 2 d y The differential equation 2 dx d 3 dx y 3 + is of order 3 and degree First Order Linear Differential Equations (Leibnitz s Linear Equations) A first order linear differential equation is of the form.a where and are functions of alone or constants. To solve A multiplying throughout by ( here is known as Integrating Factor (IF)) we get = + C Algorithm to solve a first order linear differential equation (Leibnitz s Equation) 1. Write the given equation in standard form i.e. 2. Find the integrating factor (IF) = 3. Solution is given by. IF =.IF + C C is an arbitrar Note: If the given equation is of the type then IF = and the solution is given by IF =.IF + C Example 1 Solve the differential equation: Solution: The given equation may be written as: Page 2
3 1 This is a linear differential equation of the form Where and IF = = = = Solution of 1 is given by. = ( + C = + C Example 2 Solve the differential equation: Solution: The given equation may be written as:. 1 This is a linear differential equation of the form Where and IF = = = Solution of 1 is given by. = + C = + C Example 3 Solve the differential equation: Solution: The given equation may be written as: 1 This is a linear differential equation of the form Where and IF = = = Solution of 1 is given by. = + C = + C = + C Page 3
4 Example 4 Solve the differential equation: Solution: The given equation may be written as:..1 This is a linear differential equation of the form Where and IF = = = Solution of 1 is given by. = + C C is an arbitrar Example 5 Solve the differential equation: Solution: The given equation may be written as: 1 This is a linear differential equation of the form Where and IF = = = Solution of 1 is given by. = + C. = + C. = + C = + C 10.3 Equations Reducible to Leibnitz s Equations (Bernoulli s Equations ) Differential equation of the form.b Page 4
5 where and are functions of alone or constant is called Bernoulli s equation. Dividing both sides of B by we get. Now putting B reduces to Leibnitz s equation. Example 6 Solve the differential equation: 1 Solution: The given equation may be written as: Putting 3 Using 3 in 2 we get..4 4 is a linear differential equation of the form Where and IF = = = = Solution of 4 is given by. = + C. = + C Substituting = + C = Example 7 Solve the differential equation: 1 Solution: The given equation may be written as: Putting 3 Using 3 in 2 we get..4 4 is a linear differential equation of the form Page 5
6 Where and IF = = = Solution of 4 is given by. = + C = + C = + C = + + C Substituting = + + C Example 8 Solve the differential equation: 1 Solution: The given equation may be written as: Dividing throughout by Putting 3 Using 3 in 2 we get..4 4 is a linear differential equation of the form Where and IF = = = = Solution of 4 is given by. = + C. = + C. = + C Page 6
7 Substituting = + C Example 9 Solve the differential equation: Solution: The given equation may be written as: Dividing throughout by Putting 3 Using 3 in 2 we get..4 4 is a linear differential equation of the form Where and IF = = = = Solution of 4 is given by. = + C. = + C Putting also. = + C. = + C. = + C Substituting = + C = + C is an arbitrar Exercise 10A Page 7
8 Solve the following differential equations: = + C Exact Differential Equations of First Order A differential equation of the form is said to be exact if it can be directly obtained from its primitive by differentiation. Theorem: The necessary and sufficient condition for the equation to be exact is. Working rule to solve an exact differential equation 1. For the equation check the condition for exactness i.e. 2. Solution of the given differential equation is given by Example 10 Solve the differential equation: Solution: Page 8
9 given differential equation is exact. Solution of 1 is given by: Example 11 Solve the differential equation: Solution: given differential equation is exact. Solution of 1 is given by: Example 12 Solve the differential equation: Solution: given differential equation is exact. Solution of 1 is given by: Example 13 Solve the differential equation: Solution: Page 9
10 given differential equation is exact. Solution of 1 is given by: 10.5 Equations Reducible to Exact Differential Equations Sometimes a differential equation of the form not exact i.e.. It can be made exact by multiplying the equation by some function of and known as integrating factor (IF) Integrating Factor (IF) Found By Inspection Some non-exact differential equations can be grouped or rearranged and solved directly by integration after multiplying by an integrating factor (IF) which can be found just by inspection as shown below: is Term IF Result 1. = Page 10
11 Example 14 Solve the differential equation: Solution: given differential equation is not exact. Taking as integrating factor due to presence of the term 1 may be rewritten as : Integrating 2 solution is given by : + Example 15 Solve the differential equation: Solution: Page 11
12 given differential equation is not exact. Taking as integrating factor due to presence of the term 1 may be rewritten as : Integrating 2 solution is given by : + Example 16 Solve the differential equation: Solution: given differential equation is not exact. Taking as integrating factor due to presence of the term 1 may be rewritten as : Integrating 2 solution is given by: is an arbitrary constant Example 17 Solve the differential equation: Solution: given differential equation is not exact. Rewriting 1 as Page 12
13 Taking as integrating factor due to presence of the term 2 may be rewritten as : 3 Integrating 3 solution is given by: is an arbitrar Example 18 Solve the differential equation: Rewriting 1 as: given differential equation is not exact. Integrating 2 we get the required solution as: Integrating Factor (IF) of a Non-Exact Homogeneous Equation If the equation integrating factor (IF) will be is a homogeneous equation then the provided Example 19 Solve the differential equation: Solution: given differential equation is not exact. Page 13
14 As 1 is a homogeneous equation IF = = 1 may be rewritten as : New New 2 is an exact differential equation. Solution of 2 is given by: Example 20 Solve the differential equation: Solution: given differential equation is not exact. As 1 is a homogeneous equation IF = = 1 may be rewritten after multiplying by IF as: New New 2 is an exact differential equation. Solution of 2 is given by: Page 14
15 Integrating Factor of a Non-Exact Differential Equation of the Form : If the equation is of the given form then the integrating factor (IF) will be provided Example 21 Solve the differential equation: Solution: given differential equation is not exact. As 1 is of the form IF = = 1 may be rewritten after multiplying by IF as: New New 2 is an exact differential equation. Solution of 2 is given by: Example 22 Solve the differential equation: Solution: Page 15
16 given differential equation is not exact. As 1 is of the form IF = = 1 may be rewritten after multiplying by IF as: New New 2 is an exact differential equation. Solution of 2 is given by: Integrating Factor (IF) of a Non-Exact Differential Equation shown: in which and are connected in a specific way as i. If a function of alone then IF = ii. If a function of alone then IF = Example 23 Solve the differential equation: Solution: given differential equation is not exact. Page 16
17 As 1 is neither homogeneous nor of the form Computing = Clearly say IF = 1 may be rewritten after multiplying by IF as:. New New 2 is an exact differential equation. Solution of 2 is given by: Example 24 Solve the differential equation: Solution: given differential equation is not exact. As 1 is neither homogeneous nor of the form Computing = Clearly say IF = 1 may be rewritten after multiplying by IF as: Page 17
18 New New 2 is an exact differential equation. Solution of 2 is given by: Example 25 Solve the differential equation: Solution: given differential equation is not exact. As 1 is neither homogeneous nor of the form Computing Clearly say IF = 1 may be rewritten after multiplying by IF as:. New New 2 is an exact differential equation. Solution of 2 is given by: is an arbitrar Page 18
19 Example 26 Solve the differential equation: Solution: given differential equation is not exact. As 1 is neither homogeneous nor of the form Computing Clearly say IF = 1 may be rewritten after multiplying by IF as:. New New 2 is an exact differential equation. Solution of 2 is given by: Integrating Factor (IF) of a Non-Exact Differential Equation where are constants is given by where and are connected by the relation and Example 27 Solve the differential equation: Page 19
20 Solution: given differential equation is not exact. Rewriting 1 as Comparing with standard form. and and Solving we get and IF = = 1 may be rewritten after multiplying by IF as:. New New 2 is an exact differential equation. Solution of 2 is given by: is an arbitrar Exercise 10B Solve the following differential equations: Page 20
21 Previous Years Solved Questions Q1. Solve Solution: given differential equation is not exact. Rearranging the equation as.1 Taking as integrating factor 1 may be rewritten as: Integrating 2 solution is given by : + Q2. Solve the differential equation: Page 21
22 Solution: given differential equation is not exact. As 1 is neither homogeneous nor of the form Computing Clearly say IF = 1 may be rewritten after multiplying by IF as:. New New 2 is an exact differential equation. Solution of 2 is given by: Q3. Solve is an arbitrar. Solution: given differential equation is exact. Solution is given by: Page 22
23 + Q4. Solve the differential equation: Solution: given differential equation is not exact. As 1 is a homogeneous equation IF = = 1 may be rewritten as : New New 2 is an exact differential equation. Solution of 2 is given by: Q5. Solve the differential equation: Solution: It is Bernoulli s equation which can be reduced to linear form Dividing throughout by Putting 3 Using 3 in 2 we get..4 4 is a linear differential equation of the form Page 23
24 Where and IF = = = = Solution of 4 is given by. = + C. = + C. = + C Substituting = + C = 1+ C is an arbitrar Q6. Solve the differential equation: Solution: given differential equation is not exact. As 1 is a homogeneous equation IF = = 1 may be rewritten as : New New 2 is an exact differential equation. Solution of 2 is given by: Page 24
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