ODE y''+b^2 y=0 where b is vector

In summary, an ODE, or ordinary differential equation, is a mathematical equation used to describe the relationship between a function and its derivatives. It can be solved analytically using techniques such as separation of variables, substitution, and integration. The vector b in the equation represents a parameter and can be used to model physical, biological, and economic systems. This type of ODE has applications in fields such as physics, engineering, and economics, and can be applied to real-world situations such as oscillations, waves, and population growth.
  • #1
mogul
2
0
Hi,
please help me with this task. I'm wondering what is the right result.
I have a ODE
[itex]y'' - b^2 y =0[/itex]

also the result should be
[itex]y=C e^{\pm bx}[/itex]

but what is the result when b is vector?
[itex]\vec b=(b_x, b_y)[/itex]

is this the result?
[itex]y=C e^{\pm \vec{b}x}[/itex]

or this?
[itex]y=C e^{\pm |b| x}[/itex]

how to solve it?
thanks for help!
 
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  • #2
If b is a vector, then what do you mean by "[itex]b^2[/itex]"? The dot product? Then coefficient is the square of the length of the vector.
 

Related to ODE y''+b^2 y=0 where b is vector

1. What is an ODE?

An ODE, or ordinary differential equation, is a mathematical equation that describes the relationship between a function and its derivatives. It is commonly used to model physical, biological, and economic systems.

2. How is a vector used in the ODE y''+b^2 y=0?

The vector b is used as a constant in the ODE, representing a parameter in the equation. It can be thought of as a direction and magnitude for the solution of the ODE.

3. Can this ODE be solved analytically?

Yes, this ODE can be solved analytically using techniques such as separation of variables, substitution, and integration. The resulting solution will be a function of the vector b.

4. What are the applications of this type of ODE?

This type of ODE can be used to model a variety of phenomena, including oscillations, waves, and vibrations. It is also commonly used in fields such as physics, engineering, and economics.

5. Are there any real-world examples of this ODE?

Yes, this ODE can be applied to real-world situations such as the motion of a pendulum, the behavior of a spring-mass system, and the growth of populations in ecology.

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