2.1 Introduction to DE 2.2 Concept of Solution 2.3Separation of Variable 2.4 Homogeneous Eq 2.5 Linear Eq 2.6 Exact Eq 2.7 Application of 1 st

2.1INTRODUCTION

1st order differential equation :- 2 nd order differential equation :- nth order differential equation :-

2.2CONCEPT OF SOLUTION If the solution of nth order ODE contains n arbitrary constants, then the solution is called GENERAL SOLUTION of the differential equation. A solution of nth order ODE without any arbitrary constant is called PARTICULAR SOLUTION. The functional relationship between the independent variable and the dependent variable (such as y = f(x)) which satisfies the given differential equation is called the solution of the differential equation.

2.2CONCEPT OF SOLUTION Exercises 1) Given that. Show that a) and is a solution. b), C and D are constants is also a solution. 2) If, prove that.

1 ST ORDER OF DE There are 4 types of 1 st order differential equations:- i)Separable equations ii)Homogeneous equations iii)Linear equations iv)Exact equations

2.3SEPARABLE DE Write the eqn in the form of Integrate both sides Simplify the solution

2.3SEPARABLE DE Example Solve the following differential equation: Find the particular solution of the following DE when :

2.3SEPARABLE DE Example Solve the following differential equation: Find the particular solution of the following DE :

2.3SEPARABLE DE Exercises Find the general solution of each of the following separable equations. a) b) c) d) e) f)

2.4HOMOGENEOUS EQUATIONS General form: Method of solution: 1. Show that 2. Substitute and into the general form of homogeneous equation. 3. Separate variables x and v to form separable equation. 4. Solve the separable equation. 5. Substitute into solution in step 4 and simplify the solution.

2.4HOMOGENEOUS EQUATIONS Example a: Show that the DE is the homogeneous equation Example b: Solve the DE Example c: Solve the DE

2.4HOMOGENEOUS EQUATIONS Example e:. By using and, solve. Example d: Solve the differential equation with condition y = 2 when x = 2.

2.4HOMOGENEOUS EQUATIONS Exercises Verify that each of the following equations is homogeneous and then solve it. a) b) c) d) e) f)

2.5LINEAR EQUATIONS General form:- To solve:- 1 st, arrange the eqn. to form where and 2 nd, obtain the integrating factor: 3 rd, multiply the integrating factor with 1 st eqn with to become 4 th,The eqn can be written in the form of 5 th, Simplify to Finally, the general solution of linear equation is

2.5LINEAR EQUATIONS Example d: Solve the equation Example a: Solve the equation Example b: Solve the equation Example c: Solve the equation Example e: Solve

2.5LINEAR EQUATIONS Exercises Find the general solution of the following linear equations. a) b) c) d) e) f)

2.6EXACT EQUATIONS

Solutions c)Integrate w.r.t x : e)General solution :

EXACT EQUATIONS Example : Solve the differential equation Example : Solve where 2.6EXACT EQUATIONS

Exercises 2.7 Solve the given differential equations. a) b) c) d) e) f)

2.7APPLICATIONS OF 1 ST DE – NEWTON’S LAW OF COOLING

Example 25 : A pie is removed from an oven with temperature of and placed to cool in a room with temperature of. In 15 minutes, the pie has a temperature of. Determine the time required to cool the pie to a temperature of. Example 26 : The temperature of a dead body when it was found at 3 o’clock in the morning is. The surrounding temperature at that time was. After two hours, the temperature of the dead body decreased to. Assuming that the normal body temperature is, determine the time of murdered.