bigubau said:Where did you get this problem from ? I'm almost sure there's no analytical solution (Mathematica couldn't find any combination of elementary/special functions which would satisfy the equation) to it, so an approximate solution would be the best you could ask for.
bigubau said:Yes, it looks correct. However, my guess is that the nonlinearity in the RHS spoils the <neat> intregrability.
An inhomogeneous second order ODE with non-constant coefficient is a type of differential equation that involves a second derivative, a non-constant coefficient, and a non-zero function on the right-hand side of the equation. It is a more complex type of ODE compared to a homogeneous second order ODE with constant coefficient.
The non-constant coefficient in an inhomogeneous second order ODE can make it difficult to solve because it cannot be factored out like a constant coefficient. This means that the usual methods for solving ODEs, such as separation of variables or using an integrating factor, may not work. Instead, more advanced techniques such as variation of parameters or the method of undetermined coefficients may be needed.
Inhomogeneous second order ODEs with non-constant coefficient can be used to model a variety of physical phenomena, such as the motion of a damped harmonic oscillator, the behavior of an electric circuit, or the growth of a population with a varying rate of change. They are also commonly used in engineering and physics to describe systems that involve non-linearities and external forces.
To check if your solution is correct, you can substitute it back into the original ODE and see if it satisfies the equation. You can also take the first and second derivatives of your solution and see if they match the original ODE. Additionally, you can compare your solution to known solutions or use numerical methods to approximate the solution and compare it to your own.
Some tips for solving inhomogeneous second order ODEs with non-constant coefficient include trying to simplify the equation by making a substitution or using a symmetry argument, looking for patterns in the non-homogeneous term, and being familiar with a variety of solution techniques such as variation of parameters and the method of undetermined coefficients. It is also helpful to practice solving different types of inhomogeneous second order ODEs to build your problem-solving skills.