2nd order linear homogeneous DE

In summary, when solving a 2nd order linear non-homogeneous DE where it is equal to a constant, superposition of 2 solutions occurs when the roots of the auxiliary equation are imaginary. The particular solution can be found by solving for a constant k using q(0), q'(0), q''(0). It is not always necessary to use Euler's identity in this case. In regards to another DE equal to xsinx, the solution can have derivatives with terms xsinx, xcosx, sinx, cosx by the product rule. To prove that these functions are linearly independent, one must show that their linear combinations always result in a non-zero function.
  • #1
mathnerd15
109
0
when you solve a 2nd order linear non-homogeneous DE, where it is equal to a constant as in Kirchoff's 2nd Law and the roots of the auxiliary equation are imaginary then you have superposition of 2 solutions. so the particular solution is equal to a constant k and you can solve for this by taking q(0), q'(0), q''(0) and it will have a constant form? I'd like to spend some time studying the theory of this. is it always necessary to use Euler's identity?

[tex]q(t)=\frac{-1}{25(25e^{20t})}cos(10\sqrt{46}t)+\frac{21}{50\sqrt{46}e^{20t}}sin(10 \sqrt{46}t)+\frac{1}{25}[/tex]on another question in the DE which is equal to xsinx can only have derivatives with terms xsinx, xcosx, sinx, cosx by the product rule. so in this case you guess the solution as Axsinx+Bxcosx+Csinx+Dcosx... how do you prove then that these functions are linearly independent?
 
Last edited:
Physics news on Phys.org
  • #2
By definition, a homogeneous ODE is equal to zero.
 
  • Like
Likes 1 person

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

A 2nd order linear homogeneous differential equation is a mathematical expression that includes a dependent variable, its first and second derivatives, and constant coefficients. It is considered linear if the dependent variable and its derivatives are raised to the first power and homogeneous if all terms in the equation have the same degree of the dependent variable.

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

To solve a 2nd order linear homogeneous differential equation, you must first identify the values of the constant coefficients and the dependent variable. Then, you can use various methods such as the method of undetermined coefficients or the method of variation of parameters to find the general solution.

3. 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 formula that includes arbitrary constants and satisfies the original differential equation. It can be written in the form of a linear combination of two linearly independent solutions.

4. What is the role of initial conditions in solving a 2nd order linear homogeneous differential equation?

Initial conditions are values given for the dependent variable and its derivatives at a specific point. They are used to determine the values of the arbitrary constants in the general solution and find a particular solution that satisfies both the differential equation and the given initial conditions.

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

2nd order linear homogeneous differential equations have many applications in various fields such as physics, engineering, and biology. They can be used to model oscillations, vibrations, and other physical phenomena. They are also important in understanding the behavior of electrical circuits and population dynamics.

Similar threads

I Second order non-homogeneous linear ordinary differential equation
    • Differential Equations
Replies
2
Views
2K
I Question about solving linear first order non-homogeneous ODEs
    • Differential Equations
Replies
3
Views
789
I Explanation of all "its linearly independent derivatives"
    • Differential Equations
Replies
3
Views
1K
I What exactly is linear dependency? 2nd and 3rd Order Diff EQ
    • Differential Equations
Replies
1
Views
2K
Why does the 2nd order homogeneous linear ODE have 2 general solutions?
    • Differential Equations
Replies
1
Views
2K
2nd order, linear, homogeneous proof
    • Differential Equations
Replies
4
Views
1K
Solving a 2nd order non-linear DE by dimensional analysis/observation
    • Introductory Physics Homework Help
Replies
3
Views
845
MHB Uniqueness of Solutions for 2nd Order Linear Homogeneous ODEs
    • Differential Equations
Replies
2
Views
2K
MHB Solutions of DE System & 2nd Order Differential Equation
    • Differential Equations
Replies
2
Views
1K
I Solution for 1st order, homogenous PDE
    • Differential Equations
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top