General Higher Order Linear Non-Homogeneous Diff Eq's

In summary, the general solution to higher order linear non-homogeneous differential equations is of the form ##y=y_h+y_p##, where ##y_h## is the solution to the respective homogeneous equation and ##y_p## is a particular solution of the equation. The homogeneous solution includes as many terms as possible, with a parameter of constants for each term to include various solutions. We only need one particular solution because it represents a family of solutions that satisfy the non-homogeneous differential equation. Any additional solutions can be expressed as differences between particular solutions.
  • #1
SrVishi
75
15
Hello, I am learning about the general solution to higher order linear non-homogeneous differential equations. I know that the general solution of such an equation is of the form ##y=y_h+y_p## where ##y_h## is the solution to the respective homogeneous equation and ##y_p## is a particular solution of the equation. I also realize that in the homogeneous solution, you want to include as many terms as possible, even giving a parameter of constants to each of these terms to include as many different solutions as possible. My question is, why do we only need one particular solution? If we are trying to make our general solution as "general" as possible, why do we not include a family of solutions that satisfy the non-homogeneous differential equation?
 
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  • #2
The homogeneous solutions are differences of the particular solutions.
yh=yp2-yp1
if we have
one yp and all yh
we have all solutions
say we are worried we do not have yp2
yp2=yp1+(yp2-yp1)
but we have yp1 and (yp2-yp1) is homogeneous so we have it
so we had yp2 all along
 

1. What is the general form of a higher order linear non-homogeneous differential equation?

The general form of a higher order linear non-homogeneous differential equation is yn(x) + pn-1(x)yn-1(x) + ... + p1(x)y'(x) + p0(x)y(x) = f(x), where n is the highest order of the derivative, y is the dependent variable, pi(x) are the coefficients of the derivatives, and f(x) is the non-homogeneous term.

2. What is the difference between a non-homogeneous and a homogeneous differential equation?

In a non-homogeneous differential equation, the non-homogeneous term f(x) is present, while in a homogeneous differential equation, f(x) is equal to zero. This means that the solution to a non-homogeneous differential equation will depend on the values of f(x), while the solution to a homogeneous differential equation will only depend on the initial conditions and the coefficients of the derivatives.

3. How do you solve a higher order linear non-homogeneous differential equation?

To solve a higher order linear non-homogeneous differential equation, you can use the method of undetermined coefficients or variation of parameters. The method of undetermined coefficients involves guessing a particular solution based on the form of f(x) and then finding the complementary solution using the characteristic equation. Variation of parameters involves finding a particular solution by integrating a set of functions and then adding it to the complementary solution.

4. What is the characteristic equation of a higher order linear non-homogeneous differential equation?

The characteristic equation of a higher order linear non-homogeneous differential equation is yn(x) + pn-1(x)yn-1(x) + ... + p1(x)y'(x) + p0(x)y(x) = 0. This equation is obtained by setting f(x) equal to zero and solving for the roots of the equation.

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

Higher order linear non-homogeneous differential equations can be used to model a variety of physical phenomena, such as the motion of a mass on a spring, the growth of a population over time, or the flow of electricity in a circuit. They are also commonly used in engineering, physics, and other scientific fields to describe systems that involve multiple dependent variables and their derivatives.

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