Is the ODE xy'' + siny = 0 Linear or Homogeneous?

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In summary, the given ODE is not linear because its differential operator is not linear. It also cannot be written in the form y'' + p(x)y' + q(x)y = 0, so it is not homogeneous.
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Homework Statement


Is the following ODE linear? If so, is it homogeneous?

xy'' + siny = 0, where y = y(x)


Homework Equations


Linear = coefficients of unknown function y(x) and its derivatives only depend on x, not the unknown

Homogeneous: can be written in form y'' + p(x)y' + q(x)y = 0


The Attempt at a Solution



I'm confused in that the only way to get rid of the sin from the y is to put an arcsin in front of the y'' term. Can one just 'ignore' the sin and say it is linear and also homogeneous by the definitions above?

Thanks for any help.
 
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  • #2
Write [tex]L[y] = x y'' + \sin(y)[/tex]. [tex]L[/tex] is called a differential operator (a function of a function), and the given ODE is the same as finding the y so that
[tex]L[y] = 0[/tex]

An ODE is linear if its differential operator ([tex]L[/tex] above) is linear: in other words, for any two functions [tex]y_1,y_2[/tex] and two numbers [tex]a,b[/tex] we have
[tex]L[a y_1 + b y_2] = a L[y_1] + b L[y_2] [/tex]
So the given ODE cannot be linear, since
[tex]L[a y_1 + b y_2] = axy_1'' + bxy_2'' + \sin(a y_1 + b y_2)[/tex]
and
[tex]aL[y_1] + bL[y_2] = axy_1'' + bxy_2'' + a\sin(y_1) + b\sin(y_2)[/tex]
which are not equal.

Notice that if [tex]L[y] = a_0(x) y(x) + a_1(x) y' + \ldots + a_n(x) y^{(n)}(x) [/tex] then
[tex]L[/tex] is linear by the above definition.
 

FAQ: Is the ODE xy'' + siny = 0 Linear or Homogeneous?

What is a linear ordinary differential equation (ODE)?

A linear ordinary differential equation (ODE) is a mathematical equation that involves a function and its derivatives (or differentials) with respect to one independent variable. It is called "linear" because the function and its derivatives are only multiplied by constants and added together, without any other operations such as multiplication by other functions or taking the square root.

How do you know if an ODE is linear?

An ODE is linear if it satisfies the following conditions:

  • The dependent variable and its derivatives appear to the first power only.
  • There are no products of the dependent variable and its derivatives.
  • There are no functions of the dependent variable and its derivatives, such as sine or cosine.

Can a nonlinear ODE be transformed into a linear ODE?

Yes, a nonlinear ODE can be transformed into a linear ODE by using a suitable change of variables or substitution. This involves transforming the original dependent variable into a new variable that is linearly related to the original dependent variable, and then rewriting the ODE in terms of the new variable.

What are the advantages of solving a linear ODE?

Solving a linear ODE is advantageous because it often has a closed-form solution, meaning an explicit formula for the dependent variable in terms of the independent variable. This makes it easier to analyze and understand the behavior of the system described by the ODE.

What are some common applications of linear ODEs?

Linear ODEs are used in a variety of fields, including physics, engineering, economics, and biology. They are often used to model systems that involve rates of change, such as population growth, chemical reactions, and circuit analysis. They are also used in control systems, where the goal is to manipulate the behavior of a system by controlling its inputs or outputs.

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