Nonlinear Second Order ODE: Can We Find an Analytical Solution?

In summary, the conversation discusses a nonlinear second order ODE with constants a and b, and the attempts at finding an analytical solution for it. The case where b=0 and a≠0 results in a polynomial solution of 2nd order, while the case where a=0 and b≠0 leads to the modified Bessel equation of order zero. Transformation with x=e^t is suggested, but has not been successful so far.
  • #1
tse8682
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I'm trying to solve the following nonlinear second order ODE where ##a## and ##b## are constants: $$\frac{d^2y}{dx^2}+\frac{1}{x}\frac{dy}{dx}-\frac{y}{ay+b}=0$$ It looks somewhat like the modified Bessel equation, except the third term on the left makes it nonlinear. I've been trying to determine some way to find an analytical solution but haven't been able to come up with anything. It doesn't help much but it can also be written:$$\frac{1}{x}\frac{d}{dx}\left(x\frac{dy}{dx}\right)=\frac{y}{ay+b}$$Any suggestions would be greatly appreciated, thanks!
 
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  • #2
Looks like for the case that ##b=0,a\neq 0## there is the analytical solution because then the ODE becomes linear with non constant coefficients (and I think the solution is a polynomial of 2nd order).

The case that ##a=0,b\neq 0## also seems to fallback to linear ODE as well so there should be an analytical solution.

But I am all out of ideas how to effectively treat the case ##a,b\neq 0##.
 
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  • #3
Delta2 said:
Looks like for the case that ##b=0,a\neq 0## there is the analytical solution because then the ODE becomes linear with non constant coefficients (and I think the solution is a polynomial of 2nd order).

The case that ##a=0,b\neq 0## also seems to fallback to linear ODE as well so there should be an analytical solution.

But I am all out of ideas how to effectively treat the case ##a,b\neq 0##.

Yeah, if ##b=0,a\neq 0## then the solution is ##y=\frac{x^2}{4a}+C_1\ln{x}+C_2##. If ##a=0,b\neq 0##, then it becomes the modified Bessel equation of order zero and the solution is ##y=C_1I_0\left(\frac{x}{\sqrt{b}}\right)+C_2K_0\left(\frac{x}{\sqrt{b}}\right)##.

It can be transformed if ##x=e^t## so ##t=\ln{x}##. With that it becomes: $$\frac{d^2y}{dt^2}=\frac{e^{2t}y}{ay+b}$$ They have another transformation here for equations that kinda look like that but I haven't been able to get that transformation to work.
 
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Related to Nonlinear Second Order ODE: Can We Find an Analytical Solution?

1. What is a nonlinear second order ODE?

A nonlinear second order ODE (ordinary differential equation) is a type of mathematical equation that involves a dependent variable, its derivatives, and independent variables, where the highest derivative is squared or raised to a power other than 1. This makes the equation nonlinear, meaning that it cannot be written in the form of a straight line.

2. How is a nonlinear second order ODE different from a linear second order ODE?

A linear second order ODE can be written in the form of a straight line, while a nonlinear second order ODE cannot. In a linear equation, the dependent variable and its derivatives appear only in the first power, while in a nonlinear equation, they can appear in higher powers. This makes nonlinear equations more complex and difficult to solve analytically.

3. What are some real-world applications of nonlinear second order ODEs?

Nonlinear second order ODEs are used to model many physical and natural phenomena, such as population growth, chemical reactions, and oscillating systems. They are also used in engineering and control systems to describe the behavior of complex systems.

4. How are nonlinear second order ODEs solved?

Nonlinear second order ODEs can be solved analytically using methods such as power series, perturbation, and variation of parameters. However, in most cases, they cannot be solved in closed form and require numerical methods for approximation.

5. What are some challenges in solving nonlinear second order ODEs?

One of the biggest challenges in solving nonlinear second order ODEs is finding an analytical solution. In many cases, the equations are too complex to be solved analytically, and numerical methods must be used. Additionally, the behavior of nonlinear systems can be unpredictable, making it difficult to find a general solution that applies to all cases.

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