Classify the given second-order linear PDE

In summary, there are two approaches to using the discriminant ##b^2-4ac##: the traditional approach and the textbook approach. The textbook approach involves flipping the sign of the discriminant and using ##2b## as the coefficient instead of just ##b##. Both approaches are valid and can be used interchangeably.
  • #1

chwala

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Homework Statement
Classify the second order linear pde, given by;

##U_t +2U_{tt}+3U_{xx}=0##
Relevant Equations
use of discriminant
Now i learned how to use discriminant i.e ##b^2-4ac## and in using this we have;
##b^2-4ac##=##0-(4×3×2)##=##-24<0,## therefore elliptic.

The textbook has a slight different approach, which i am not familiar with as i was trained to use the discriminant at my undergraduate studies...
see textbook approach here;

1638669547425.png


and the textbook solution is here;
1638669780842.png
and i also checked wikipedia on this and they have used the discriminant approach. See the approach here; ( i am comfortable with this approach)
1638669617986.png


Both approaches should be fine, one can use either approach...correct?
 
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  • #2
The textbook is the same thing with some math modifications

1.) Instead of ##b^2-4ac## they have ##4ac-b^2## so the sign is flipped

2.) They write ##2b## as the coefficient instead of ##b##. So given that the coefficient of the cross term is called ##2b##, their discriminant becomes
##4ac-(2b)^2=4(ac-b^2)##. Since 4 is positive, the sign of the discriminant is determined by ##ac-b^2##.
 
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  • #3
Office_Shredder said:
The textbook is the same thing with some math modifications

1.) Instead of ##b^2-4ac## they have ##4ac-b^2## so the sign is flipped

2.) They write ##2b## as the coefficient instead of ##b##. So given that the coefficient of the cross term is called ##2b##, their discriminant becomes
##4ac-(2b)^2=4(ac-b^2)##. Since 4 is positive, the sign of the discriminant is determined by ##ac-b^2##.
ooooh, Thanks am trying to refresh on pde's cheers mate...:cool:
 

1. What is a second-order linear PDE?

A second-order linear PDE (partial differential equation) is a mathematical equation that involves partial derivatives of a function of two or more independent variables. It is considered linear if the dependent variable and its derivatives appear only in the first or second power and are not multiplied together. In other words, the equation is linear in its partial derivatives.

2. How do you classify a second-order linear PDE?

A second-order linear PDE can be classified based on its coefficients and the type of equation it represents. The coefficients are the numbers that appear in front of the derivatives, and they determine the type of equation (elliptic, parabolic, or hyperbolic). The type of equation is also determined by the highest-order derivatives present in the equation.

3. What is the difference between an elliptic, parabolic, and hyperbolic PDE?

An elliptic PDE has constant coefficients and represents a steady-state or equilibrium situation, such as the heat equation. A parabolic PDE has one variable with a coefficient of 1 and represents a time-dependent situation, such as the diffusion equation. A hyperbolic PDE has two variables with coefficients of opposite signs and represents a wave-like behavior, such as the wave equation.

4. How do you solve a second-order linear PDE?

The method for solving a second-order linear PDE depends on its type. Elliptic PDEs can be solved using techniques such as separation of variables, while parabolic and hyperbolic PDEs require initial or boundary conditions to be specified in order to find a unique solution. Numerical methods, such as finite difference or finite element methods, can also be used to solve PDEs.

5. What are some real-world applications of second-order linear PDEs?

Second-order linear PDEs are used to model a variety of physical phenomena, such as heat transfer, diffusion, and wave propagation. They are commonly used in engineering, physics, and other fields to solve problems related to fluid dynamics, electromagnetism, and structural mechanics. Examples of real-world applications include predicting the temperature distribution in a metal rod, modeling the spread of pollutants in the atmosphere, and analyzing the behavior of sound waves in a room.

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