MTH401 Handouts

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

dy dx x

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 y=f(x)

f(x,y)=0

z=f(x,y)

„ Reverse of Differentiation „ By parts „ By substitution „ By Partial Fractions „ Reduction Formula Frequently required „ Standard Differentiation formulae „ Standard Integration Formulae

„ Something New „ Mostly old stuff Presented differently Analyzed differently Applied Differently

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

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

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

Specific Examples of ODE’s