Ordinary Differential Equations

[arepeace]

I am just curious, are these Ordinary Differential Equations exist?
1. First order nonlinear homogeneous equations
2. First order nonlinear non-homogeneous equations
3. Second order nonlinear homogeneous equations
4. Second order nonlinear non-homogeneous equations

If any of those equations exist, can you give me an example for each category, and method of the solution, if possible? I am really interested to know. Thank you so much for your attention.

 

[Deleted member 4993]

I am just curious, are these Ordinary Differential Equations exist?
1. First order nonlinear homogeneous equations
2. First order nonlinear non-homogeneous equations
3. Second order nonlinear homogeneous equations
4. Second order nonlinear non-homogeneous equations

If any of those equations exist, can you give me an example for each category, and method of the solution, if possible? I am really interested to know. Thank you so much for your attention.

What are your thoughts?

Please share your work with us ...even if you know it is wrong

If you are stuck at the beginning tell us and we'll start with the definitions.

You need to read the rules of this forum. Please read the post titled "Read before Posting" at the following URL:

http://www.freemathhelp.com/forum/th...Before-Posting

 

[Ishuda]

I am just curious, are these Ordinary Differential Equations exist?
1. First order nonlinear homogeneous equations
2. First order nonlinear non-homogeneous equations
3. Second order nonlinear homogeneous equations
4. Second order nonlinear non-homogeneous equations

If any of those equations exist, can you give me an example for each category, and method of the solution, if possible? I am really interested to know. Thank you so much for your attention.

Just to emphasize the "If you are stuck at the beginning tell us and we'll start with the definitions." from the post by Subhotosh Khan: What determines the order of a differential equation? What determines if a differential equation is non-linear.

 

[arepeace]

Dear Subhotosh Khan & Ishuda,

Here are my thoughts:
Linearity of ODE determines by the equation form. ODE is linear if the equation can be written as:

a_n(x)y⁽ⁿ⁾ + ... + a₂(x)y'' + a₁(x)y' + a₀(x)y = b(x) -----------(1)

where coefficients of a’s and b could be functions of zero, constant, and linear or nonlinear functions of x. Otherwise is nonlinear ODE.
So, I think the equation form for nonlinear ODE is maybe like this:

a_n(x,y)y⁽ⁿ⁾ + ... + a₂(x,y)y'' + a₁(x,y)y' + a₀(x)y = b(x,y) -------------(2)

where coefficients of a_{n, ...,} a₂, a1 and b are the same function as in linear ODE, except there is additional linear or nonlinear functions of y.

To determine homogeneous or non-homogeneous ODE:
For first order linear ODE, homogeneous equation can be written as:

y' = f(x,y) = F(y/x)

This means every variable x and y present must be able to be written together as y/x, otherwise the equation is non-homogeneous. By transforming the equation into separable equation, the equation is solvable.

For second order linear ODE, homogeneous equation can be written as:

a₂(x)y'' + a₁(x)y' + a₀(x)y = 0

If coefficients a’s are 0 (a₂ can't be 0) or constant numbers, then the equation is solvable.

But here I have additional question: if coefficients a’s are linear or nonlinear functions of x, is there any way to find the possible solution? I know that generally, it is not easy to discover the particular solutions. However, I found Cauchy-Euler equation can solved second order ODE with form:

ax²y'' + bxy' + cy = b(x)

where a, b and c are constant and b(x) could be zero, constant, linear or nonlinear functions of x.
Is there any equation similar as Cauchy-Euler equation?

My thoughts about my earliest questions: are these ODE exist?
1. First order nonlinear homogeneous equations
Referring equation (2): for nonlinear ODE:

a₁(x,y)y' + a₀(x)y = b(x,y)

so y' = [-a₀(x)y + b(x,y)]/a₁(x,y) --------------(3)

By considering RHS of equation, I think this equation can be homogeneous. By transforming the equation into separable equation, the equation is solvable. (I think I have answered the question)

2. First order nonlinear non-homogeneous equations
I realize equation (3) can also be non-homogeneous. But I don’t know how to solve it, as I can’t make the equation separable and I can’t use integrating factor. Maybe it is unsolvable?

3. Second order nonlinear homogeneous equations
Referring equation (2):

a₂(x,y)y'' + a₁(x,y)y' + a₀(x)y = 0

4. Second order nonlinear non-homogeneous equations
Referring equation (2):

a₂(x,y)y'' + a₁(x,y)y' + a₀(x)y = b(x,y)

For second order nonlinear homogeneous or non-homogeneous equations, I have no idea how to solve them. Maybe they are unsolvable, I guess.

 

[Ishuda]

Your definition for linear and non-linear ODEs are somewhat restrictive. A simple definition for non-linearity of an ODE is that neither the function being solved for (y in your examples) nor its derivatives have an exponent associate with them other than 1 [or, equivalently, have an exponent other that one] nor do they occur in nonlinear functions. Thus, look at a general form
$\displaystyle \underset{n=0}{\overset{N}{\Sigma}}\, a_n(x)\, f_n\boldsymbol{(}\frac{d^ny}{dx^n}\boldsymbol{)}\, =\, b(x)$
The order is the largest such m such that a_m(x) is not zero. If all of the functions f_n with non-zero coefficients are linear (of the form f_n(t) = c_n t + d_n where c_n and d_n are constant) then the ODE is linear, otherwise it is non-linear. Note that if any d_n is not zero, it can just be collected into the b(x), so we can assume all of the d_n are zero. Given that, if b(x) is zero the ODE is homogeneous, otherwise it is non-homogeneous.

 

[arepeace]

That general form - I have never seen it before. Now I realize my ODE definition is restrictive. Thank you.