Second order differential equation

In summary, The conversation discusses how to show that y1(x) = e^(2+i)x and y2(x) = e^(2-i)x, i=sqrt(-1) are two linearly independent functions and how to obtain a second order linear differential equation with constant coefficients using these two functions as its fundamental solutions. The first part involves using the definition of the Wronskian and showing that it is not equal to zero. The second part involves solving for the auxiliary equation by setting (r+2+i)(r+2-i), and then using that to check if the two original functions satisfy the differential equation.
  • #1
bobey
32
0
show that y1(x) = e^(2+i)x and y2(x) = e^(2-i)x, i=sqrt(-1) are two linearly independent functions

hence obtain a second order linear differential equation with constant coefficients each that y1(x) and y2(x) are its two fundamental solutions.

my attempt :

for the first part, I use the definition of wroskian = y1y2'-y2y1' and show it not equal to zero... ok

the second part, I don't know how to do it...

how to get the second order differential equation?

is that setting : (r+2+i)(r+2-i) to get the auxillary equation? is that possible? can someone show me to solve this problem?:confused:
 
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  • #2
If you want that auxiliary equation, what will be the differential equation? When you get it, you can check whether the two original functions satisfy it.
 
  • #3
how to do that... still blurr...
 

1. What is a Second Order Differential Equation?

A second order differential equation is a mathematical equation that involves the second derivative of a function. It is used to model a variety of physical phenomena in fields such as physics, engineering, and economics.

2. How is a Second Order Differential Equation different from a First Order Differential Equation?

A second order differential equation involves the second derivative, while a first order differential equation only involves the first derivative. This means that a second order differential equation has a higher degree of complexity and can represent more complex physical phenomena.

3. What is the general form of a Second Order Differential Equation?

The general form of a second order differential equation is: y'' = f(x,y,y'), where y is the dependent variable, x is the independent variable, and f(x,y,y') is a function that represents the relationship between y, x, and their derivatives.

4. How do you solve a Second Order Differential Equation?

The method for solving a second order differential equation depends on the type of equation. Some common techniques include separation of variables, substitution, and using an integrating factor. In some cases, a numerical or computer-based approach may be necessary.

5. What are some real-world applications of Second Order Differential Equations?

Second order differential equations are used to model a wide range of phenomena, including motion of objects under the influence of forces, electrical circuits, and population dynamics. They are also commonly used in engineering for solving problems related to vibrations, heat transfer, and fluid flow.

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