ODE Syllabus

The syllabus for the Ordinary Differential Equations being taught to the first-year B.Tech. students:

Ordinary Differential Equations of First Order: Nature of ordinary differential equations, Modeling engineering systems as differential equations, Solution methods, Applications to  law of natural  growth and decay problems, Newton’s  law of cooling,  Chemical Reactions and Solutions, Orthogonal Trajectories, Linear Equations and Non- Linear Equations, R-L circuits with step unit,  R-L Circuits with input.

Ordinary Differential Equations of Second Order: Formulation, Modeling engineering systems as second order ODEs, Conversion of some models to Differential Equations of second and higher order, Homogeneous equations (Complementary Function), Non-Homogeneous equations (Particular Integral).

Applications of Second and Higher Order ODEs: Cauchy’s linear equation, Legendre’s linear equation, Method of variation of parameters, Solving system of simultaneous ODEs. L-C-R Circuits with and without e.m.f., Oscillations of a system with damping or forcing, Oscillation and deflection of beams.

Laplace Transforms: Laplace transforms of some standard functions, properties of Laplace transforms - Linearity, First Shifting Property, Change of Scale Property, Transforms of derivatives & integrals, multiplication by $t^n$, division by $t$, Inverse  Laplace transforms,  Convolution theorem, transforms of periodic functions and Unit-Step function. Applications: Solving ODE using Laplace transforms method.

Matrices: Linear independence and dependence of a set of vectors,  Eigenvalues and eigenvectors, Stability of a system of ODEs by eigenvalues,  Orthogonality of  eigenvectors, Complex matrices, Quadratic forms and canonical forms, Diagonalization

Text Books:

1.    Advanced Engineering Mathematics by Erwin Kreyszig, Wiley India, 8^th Edition, 2006.

2.    Advanced Engineering Mathematics by Michael D. Greenberg, Pearson Education, 4^th Edition, 2008.

References:

1.    Advanced Engineering Mathematics by Alan Jeffrey, Elsevier, 2010.

2.    Higher Engineering Mathematics by B.V. Ramana, McGraw Hill Co., 2010.

3.    Engineering Mathematics, by Anthony Croft, Robert Davison, Martin Hargreaves, Pearson Education, 3^rd Edition, 2009

The First Quiz

Students of the Ordinary Differential Equations course had their first quiz on Monday, August 29, 2011 - exactly one week into their first semester. There was some consternation till it was pointed out that an early quiz is necessarily an easy quiz.

Here are the questions and their solutions:

1. Give the order and degree of $ 2x^2y^{\prime\prime} -3y^\prime +y=0$.
   
   Solution: The highest order derivative present is of second order ($y^{\prime\prime}$), so the ODE has order 2. The highest order derivative is present with degree 1, so the ODE has degree 1.
   
   
2. 
   
   
   1. Verify that $y(x)= ce^{-x}+2$ is a general solution to $y^\prime +y =2$.
      
      Solution: We calculate $y^\prime = -ce^{-x}$. Now we substitute $y$ and $y^\prime$ into the ODE:
      \[ LHS = y^\prime +y = -ce^{-x} + ce^{-x}+2 = 2 = RHS\]
      So the ODE is satisfied and the given $y$ is a general solution.
   2. Find the particular solution to this ODE given that $y=3.2$ when $x=0$.
      
      Solution: Substitute this pair of values into the general solution:
      \[ce^{-0} + 2 = 3.2\]
      We solve this for $c$ and get $c=1.2$. So the particular solution for the given initial condition is $y= 1.2 e^{-x}+2$.
      
   3. Graph this solution.
      
      Solution: