Why does the 2nd order homogeneous linear ODE have 2 general solutions?

In summary, 2nd order linear homogeneous ODEs have two general solutions because all such equations can be solved by the characteristic equation. This is due to the fact that the characteristic equation can only have two solutions, leading to the two general solutions for the ODE. This logic holds for all 2nd order linear homogeneous ODEs, making it the only case to solve the question.
  • #1
kidsasd987
143
4
why not the 2nd order linear homogeneous ODEs have three Linearly independent solutions or more? I know for the characteristic equation, we can only find 2 answers but.. just wondering if that is the only case to solve the question and if it is, then why it has to be.

so my question is,1. 2nd order linear homogeneous ODE has 2 general solutions. but why?

2. derivation?
 
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  • #2
is that because all 2nd order linear homogeneous ODEs can be solved by characteristic equation?

so the logic flow here is, 1. all 2nd order linear homogeneous ODES can be solved by characteristic eqs. 2. therefore we have 2 solutions.

not the reverse.
 

Related to Why does the 2nd order homogeneous linear ODE have 2 general solutions?

1. Why does the 2nd order homogeneous linear ODE have 2 general solutions?

The 2nd order homogeneous linear ODE has 2 general solutions because it is a linear equation, which means that it can be expressed as a linear combination of its solutions. In other words, the general solution is a combination of two linearly independent solutions, which results in two possible solutions.

2. What is the difference between a particular solution and a general solution?

A particular solution is a specific solution that satisfies the given initial conditions of the ODE, while a general solution is a family of solutions that satisfy the ODE without any initial conditions. In other words, the general solution includes all possible solutions of the ODE, while a particular solution is just one specific solution.

3. How can I determine the two general solutions of a 2nd order homogeneous linear ODE?

To determine the two general solutions of a 2nd order homogeneous linear ODE, you can use the characteristic equation method. This involves finding the roots of the characteristic equation and using those roots to find the two linearly independent solutions. Another method is using the method of undetermined coefficients, which involves guessing the form of the two solutions and solving for the coefficients.

4. Can the two general solutions of a 2nd order homogeneous linear ODE be equal?

No, the two general solutions of a 2nd order homogeneous linear ODE cannot be equal. This is because the solutions must be linearly independent, meaning that they cannot be multiples of each other. If the two solutions were equal, it would mean that they are linearly dependent, which is not possible for a general solution of a 2nd order homogeneous linear ODE.

5. How can I use the two general solutions to find a particular solution of a 2nd order homogeneous linear ODE?

You can use the two general solutions to find a particular solution of a 2nd order homogeneous linear ODE by using the method of variation of parameters. This involves finding the Wronskian of the two solutions and using it to find the particular solution. Alternatively, you can also use the initial conditions to determine the values of the arbitrary constants in the general solution, resulting in a particular solution.

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