Help simple 2nd order linear ODE

In summary, The conversation discusses the method for solving a specific ODE with constant coefficients. The speaker is struggling with the nonconstant case and asks for a hint. The answer provided by Wolfram Alpha involves using a specific substitution to simplify the equation. Depending on the sign of the constant, the solutions can be expressed as sums of exponential or sinusoidal functions of the square root of the constant multiplied by t.
  • #1
y2kevin
6
0
Hi, could anyone give me a hint on what method to use to solve this ODE:

v''+(2/t)v'+(b)v=0,

b is a constant and v=v(t).

Most of my ODE training resolves around how to solve the above equation with constant coefficients. AND all of my reference books say that "the nonconstant case is difficult and you won't run into it." Well, here I am, stuck.

The answer wolfram alpha gave is:

v(r) = (c_1 e^(-sqrt(-k^2) r))/r+(c_2 e^(sqrt(-k^2) r))/(2 sqrt(-k^2) r)

No idea on how it got it though.
 
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  • #2
Let v = y/t
which turns the ODE to :
y'' + b y = 0
So, depending the sign of b, the solutions y(t) are sums of exponential or sinusoidal functions of (sqrt(-b)*t) or of (sqrt(b)*t)
 

1. What is a 2nd order linear ODE?

A 2nd order linear ODE (ordinary differential equation) is a mathematical equation that describes the relationship between a function and its derivatives. It contains a dependent variable, its first and second derivatives, and may also include independent variables and constants.

2. How do you solve a 2nd order linear ODE?

To solve a 2nd order linear ODE, you can use various methods such as the method of undetermined coefficients, variation of parameters, or transforming it into a first order system. The specific method used depends on the form of the equation and any initial or boundary conditions given.

3. What is the general solution of a 2nd order linear ODE?

The general solution of a 2nd order linear ODE is a solution that contains two arbitrary constants. These constants can be determined by applying any initial or boundary conditions given in the problem. The general solution can also be written in terms of a particular solution and the complementary function.

4. What is the difference between a homogeneous and non-homogeneous 2nd order linear ODE?

A homogeneous 2nd order linear ODE has a right-hand side of 0, meaning that the dependent variable and its derivatives are the only terms present. A non-homogeneous 2nd order linear ODE has a non-zero right-hand side, which may include constants or functions of the independent variable.

5. What are some real-world applications of 2nd order linear ODEs?

2nd order linear ODEs have many applications in fields such as physics, engineering, and economics. For example, they can be used to model the motion of a spring, the growth of a population, or the flow of electricity in a circuit. They can also be used to analyze the behavior of systems in various industries, such as in the design of bridges or control systems for airplanes.

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