2nd Order Linear Diff. Eqn (homogeneous)

In summary, the conversation discusses a differential equation with a non-zero g(t) term and the attempt to prove that y = c*x(t) is not a solution. It is established that g(t) being non-zero implies the equation is non-homogeneous. By substituting y = c*x(t) into the equation, it is shown that the resulting expression is different from the original equation, indicating that y = c*x(t) is not a solution.
  • #1
aznkid310
109
1

Homework Statement


Show that id y = x(t) is a solution of the diff. eqn. y'' + p(t)y' + q(t)y = g(t), where g(t) is not always zero, then y = c*x(t), where c is any constant other than 1, is not a solution.


Homework Equations


Can someone help me get started?
Also, since g(t) is not zero, this means that the equation is nonhomogeneous?


The Attempt at a Solution

 
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  • #2
Yes, this implies the DE is non-homogenous. To show that y = c*x(t) is not a solution just substitute that into the DE. Do you still get g(t) on the RHS?
 
  • #3
For y = c*x(t): x''(t) + p(t)*x'(t) + q(t)*x(t) = g(t)

What next?
 
  • #4
Note that it is given that y=x(t) is a solution, that means x''(t) + p(t)x'(t) + q(t)x(t) = g(t). The expression you get when you substitute y = cx(t) into the DE is clearly different from this. What does that tell you?
 
  • #5
aznkid310 said:
For y = c*x(t): x''(t) + p(t)*x'(t) + q(t)*x(t) = g(t)

What next?
What happened to the c?? If x''(t) + p(t)*x'(t) + q(t)*x(t) = g(t) what is
(c*x)'' + p(t)*(c*x)' + q(t)*(c*x)?
 

1. What is a 2nd order linear differential equation?

A 2nd order linear differential equation is a mathematical equation that involves a second derivative of an unknown function, along with the function itself and its first derivative. It is called linear because the unknown function and its derivatives appear only in a linear form.

2. What does it mean for a 2nd order linear differential equation to be homogeneous?

A 2nd order linear differential equation is considered homogeneous if all of its terms involve only the unknown function and its derivatives. In other words, there are no external or non-homogeneous factors affecting the equation.

3. How do you solve a 2nd order linear homogeneous differential equation?

To solve a 2nd order linear homogeneous differential equation, you can use the characteristic equation method or the method of undetermined coefficients. The characteristic equation method involves finding the roots of the characteristic equation and using them to form the general solution. The method of undetermined coefficients involves guessing a particular solution based on the form of the non-homogeneous terms in the equation.

4. What is the general solution of a 2nd order linear homogeneous differential equation?

The general solution of a 2nd order linear homogeneous differential equation is a combination of two linearly independent solutions, which are solutions that cannot be expressed as a multiple of each other. This general solution can be found using the characteristic equation method or the method of undetermined coefficients.

5. What are some real-world applications of 2nd order linear homogeneous differential equations?

2nd order linear homogeneous differential equations are commonly used in physics and engineering to model systems that involve acceleration, such as oscillating systems and circuits with inductors and capacitors. They are also used in population dynamics to model the growth or decline of a population over time.

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