MTH401 Official 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  1 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 dx 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

dy  5 y  1 dx  0

d 2 y dx2  dy   5  4 y  ex

u  v  0 y x x u  y v  u x y 2u  2u  u

x2 t 2 2 t dx

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

dy  5 y  1 dx  0

d 2 y dx2  dy   5  4 y  ex

u  v  0 y x x u  y v  u x y 2u  2u  u

x2 t 2 2 t

Specific Examples of ODE’s