1. Solutions of Second Order ODEs with CC
P M V Subbarao
Professor
Mechanical Engineering Department
I I T Delhi
Methods to find Two Independent Solutions….

2. Methods For Finding Two Linearly Independent Solutions
Characteristic (Auxiliary) Equation
Reduction of order
Variable Coefficients (Cauchy-Euler)

3. Characteristic Equation Method
This method is used only when the coefficients 2 , 1 and 0 are real constants.
Then the equation is called as homogeneous second order ODE with constant coefficients.
It’s not hard to think of some likely candidates for particular solutions of above Equation.
A simple exponential function is a natural ansatz for above ODE, due to its superior differential properties.
An educated guess for the solution is y = ex, where  is a constant to be determined.

4. Linear (Constant Coefficient) Homogeneous ODEs of Second Order OrderIf
is an ansatz, then
&
Substitute these in the Operator.
Our ansatz has thus converted a differential equation into an algebraic equation.
This is called as Characteristic Equation

5. Roots of Characteristic Equation
Using the quadratic formula, the two solutions of the characteristic equation are given by
There are three cases to consider:
(1) if 12 -42 0 > 0, then the two roots are distinct and real;
(2) if 12 -42 0 < 0, then the two roots are distinct and complex conjugates of each other;
(3) if 12 -42 0 = 0, then the two roots are degenerate and there is only one real root.
All these three cases must be studied in the context of thermofluids.

6. Real & Distinct roots Both the roots are real for
+  - are real and distinct roots.
Two independent solutions are:
The general solution to can be written as a linear superposition of the two solutions; that is,
The unknown constants c1 and c2 can be determined by the given initial conditions.

7. Complex Conjugate, Distinct roots
+  - are complex conjugate roots for

8. Euler’s Trigonometric Formula
The central mathematical fact that is of engineering interest in this context is generally called “Euler's formula”.
Euler's formula is ubiquitous in mathematics, physics, and engineering.
The physicist Richard Feynman called the equation "our jewel" and "the most remarkable formula in mathematics".
Around 1740 Euler turned his attention to the exponential function in place of logarithms and obtained the formula used today that is named after him.
It was published in 1748, obtained by comparing the series expansions of the exponential and trigonometric expressions.

9. Eulerization of Complex conjugate, distinct roots
Euler Formulae:
The unknown constants k1 and k2 can be determined by the given initial conditions.

10. Repeated roots The degenerate root is then given by
yielding only a single solution to the ode:
To satisfy two initial conditions, a second independent solution must be found separately.
One such method is Reduction of order.

12. Wright Brother’s Art of Learning Aerodynamics
At the end of their 1901 wind tunnel tests, the Wright brothers had the most detailed data in the world for the design of aircraft wings.
In 1902, they returned to Kitty Hawk with a new aircraft based on their new data.
This aircraft performed much better than the 1901 aircraft and lead directly to the successful 1903 flyer.
Results of the wind tunnel tests were also used in the design of their propellers.

13. Formulation of ODE for Design of Pitot-Static Tube Manometer : Assumptions
The fluid is assumed to be incompressible the total length of the fluid column remains fixed at L.
Assume that the probe is initially in the equilibrium position.
The pressure difference Δp is suddenly applied across it.
The fluid column will move during time t > 0.

14. The forces that are acting on the length L of the fluid
Force disturbing the equilibrium
Inertial Force
Forces opposing the change:
a. Weight of column of fluid
b. Fluid friction due to viscosity of the fluid :
The velocity of the fluid column is expected to be small and the laminar assumption is thus valid.
The viscous force opposing the motion is calculated based on the assumption of fully developed Hagen-Poiseuelle flow.
The fricitional pressure drop

15. The forces that are acting on the length L of the fluid
Newton’s Law of Motion

16. A Second Order ODE

17. A Second Order ODE with constant coefficients
Primary design criteria: ???