MTH303 Handouts

Table of Contents

Lecture No. 1 Introduction.……………………………… ..................................................3 Lecture No. 2 Fundamentals of Differential Equation.................................................5 Lecture No. 3 Separable Equations...................................................................................9 Lecture No. 4 Homogeneous Differential Equations...................................................18 Lecture No. 5 Exact Differential Equations….……......................................................26 Lecture No. 6 Integrating Factor Technique….……...................................................34 Lecture No. 7 First Order Linear Equations................................................................44 Lecture No. 8 Bernoulli Equations..................................................................................51 Lecture No. 9 Practice Examples.....................................................................................62 Lecture No. 10 Applications of First Order Differential Equations.........................78 Lecture No. 11 Radioactive Decay....................................................................................89 Lecture No. 12 Application of Non Linear Equations..................................................95 Lecture No. 13 Higher Order Linear Differential Equations...................................103 Lecture No. 14 Solutions of Higher Order Linear Equations...................................115 Lecture No. 15 Construction of a Second Solution......................................................127 Lecture No. 16 Homogeneous Linear Equations with Constant Coefficients……........134

Lecture No. 17 Method of Undetermined Coefficients

Superposition Approach…………………….……………......145 Lecture No. 18 Undetermined Coefficient Annihilator Operator Approach…..……..159 Lecture No. 19 Undetermined Coefficients Annihilator Operator Approach……......170 Lecture No. 20 Variation of Parameters.......................................................................180

Lecture 1

Background Linear y=mx+c Quadratic ax2+bx+c=0 Cubic ax3+bx2+cx+d=0 Systems of Linear equations

ax+by+c=0 lx+my+n=0

Solution ?

Equation

Differential Operator

Taking anti derivative on both sides y=ln x

From the past

 Algebra  Trigonometry  Calculus  Differentiation

 Differentiation Algebraic Functions Trigonometric Functions Logarithmic Functions Exponential Functions Inverse Trigonometric Functions

 More Differentiation Successive Differentiation Higher Order Leibnitz Theorem  Applications Maxima and Minima Tangent and Normal  Partial Derivatives dy dx x

Lecture 2

Fundamentals  Definition of a differential equation.

 Classification of differential equations.

 Solution of a differential equation.

 Initial value problems associated to DE.

 Existence and uniqueness of solutions Elements of the Theory  Applicable to: • Chemistry • Physics • Engineering • Medicine • Biology • Anthropology  Differential Equation – involves an unknown function with one or more of its derivatives  Ordinary D.E. – a function where the unknown is dependent upon only one independent variable Examples of DEs

Specific Examples of ODE’s x dy y dx y x dx xdy

d y dy y e dx dx u v y x u v x y u x y u u u x t t