Differential Equations MTH 242 Lecture 02 Dr Manshoor

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Differential Equations MTH 242 Lecture # 02 Dr. Manshoor Ahmed

Differential Equations MTH 242 Lecture # 02 Dr. Manshoor Ahmed

Summary (Recall) • Motivation and introduction. • What is a differential equation? • Classification

Summary (Recall) • Motivation and introduction. • What is a differential equation? • Classification by types. • Classification by order. • Degree of the DE. • Classification by linearity.

Solution of an ODE The interval I in definition is called as the interval

Solution of an ODE The interval I in definition is called as the interval of definition/. existence/ validity/ the domain of the solution.

Verification of a Solution

Verification of a Solution

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General Solution of an ODE

General Solution of an ODE

Particular Solution of an ODE

Particular Solution of an ODE

Singular Solution of an ODE

Singular Solution of an ODE

Families of Solution Note: This means that a single differential equation can possess an

Families of Solution Note: This means that a single differential equation can possess an infinite number of solutions corresponding to the unlimited number of choices for the parameter(s).

Problem Solution

Problem Solution

Problem Solution

Problem Solution

Practice Questions In the following problems verify that the indicated function is an explicit

Practice Questions In the following problems verify that the indicated function is an explicit solution of the given differential equation.

Initial Value Problems The initial value problems arise in physical system, the n-th order

Initial Value Problems The initial value problems arise in physical system, the n-th order Initial Value Problem.

First and second order Initial Value Problems

First and second order Initial Value Problems

Existence and Uniqueness of the solution

Existence and Uniqueness of the solution

Important Remark

Important Remark

Problem solution

Problem solution

Summary • Solution of differential equation. • Interval of the solution. • Trivial, general,

Summary • Solution of differential equation. • Interval of the solution. • Trivial, general, particular and singular Solutions. • Families of solution. • Initial Value Problems. • Existence and uniqueness theorem