ODE Course Statistics

Here are some summary data for the ODE course taught in the last semester:

(Provisional) Grade Distribution:

A	A-	B	B-	C	C-	D	F
34	35	64	55	31	17	12	6

Histogram of Final Exam Marks (%):



Final Exam Mean = 59%

Final Exam Median = 60%

Standard Deviation =  17%

Interquartile Range = 24%

ODE Final Exam

The Final Exam for ODE will be based on the following sections of the 8th edition of Kreyszig's Advanced Engineering Mathematics:

Ch 1: All, except sections 1.2 and 1.9.
Ch 2: All, except sections 2.4 and 2.13--2.15.
Ch 3: 3.0 to 3.3. From 3.6, only method of undetermined coefficients when RHS has no term which is a solution of the homogeneous equation.
Ch 5: 5.1 to 5.5.

The marks distribution will be Ch 1 (20), Ch 2 (20), Ch 3 (20), Ch 5 (40).

Further information may follow.

ODE - Nov 4 Exam - Practice Problems

From Kreyszig, 8th Edition:

Problem Set 2.2 – Q13, Q17, Q22

Problem Set 2.3 – Q8, Q16, Q20

Problem Set 2.6 – Q4, Q6, Q14, Q16

Problem Set 2.7 – Q4, Q6, Q8, Q10, Q12

Problem Set 2.8 – Q4, Q6, Q8

Problem Set 2.9 – Q13, Q15, Q17, Q21, Q22

Problem Set 2.10 – Q1, Q3, Q5, Q11, Q13

Problem Set 2.11 – Q1, Q7, Q13

Problem Set 2.12 – Q5, Q7, Q11, Q13, Q14

ODE Assignment 1 Solutions

Solutions to some of the ODE first assignment problems. Scroll through the embedded document using the controls at its base.

Assignment 1

Syllabus for ODE 1st midterm

The first midterm for the ODE course will cover the first chapter of Advanced Engineering Mathematics by Erwin Kreyszig (8th Edition). However, Sections 1.2 and 1.9 are excluded.

ODE - Assignment 1

Due Date: September 9 September 12

1. A tank contains 800 gal of water in which 200 lb of salt is dissolved. Two gallons of fresh water runs in per minute and 2 gal of the mixture in the tank, kept uniform by stirring, runs out per minute.
   
   1. Form a differential equation for the amount of salt in the tank as a function of time.
   2. How much salt is left in the tank after 5 hours?
2. A thermometer, reading $10^\circ\mathrm{C}$, is brought into a room whose temperature is $23^\circ\mathrm{C}$. Two minutes later the thermometer reading is $18^\circ\mathrm{C}$.
   
   1. Use Newton's Law of Cooling to model the change of the thermometer reading with time.
   2. How long will it take until the reading is practically $23^\circ\mathrm{C}$, say, $22.8^\circ\mathrm{C}$?
3. Solve the following initial value problem (IVP) and make a reasonably accurate sketch of the solution:
   \[ 2y^\prime + y^3 = 0;\quad y(0)=1 \]
   
4. Solve the following ODEs:
   
   1. $\left[\sin(y)\cos(y)+x\cos^2(y)\right]\,dx + x\,dy=0$
   2. $e^y\,\left[\sinh(x)\,dx + \cosh(x)\,dy\right]=0$
   3. $(x^2+y^2)\,dx - 2xy\,dy =0$
   4. $\left(\cos(xy)+\dfrac{x}{y}\right)\,dx + \left(1 + \dfrac{x}{y}\cos(xy)\right)\,dy=0$
5. 1. Under what conditions for the constants $A$,$B$,$C$,$D$, is the following ODE exact?
      \[(Ax+By)\,dx + (Cx+Dy)\,dy=0\]
   2. Solve this exact ODE.
6. 1. Solve the ordinary differential equation: $y^\prime\tan(x)=2y-8$.
   2. Sketch the family of solutions given by the general solution to the above ODE.
   3. Give the particular solution to the above ODE such that $y=0$ when $x=\pi/2$.

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: