From this formulation it is quite difficult to obtain a closed-form solution for that ODE. I'd try a change z = some function of y, but I don't expect a solution to appear easily.

Instead, I'd suggest a numerical approach if the context allows you to do so. Finite differences methods are a good point to start with.

Domingo, you may use iterations y_n''(t)+a y_n'(t)+c y(t)=A sin(wt)-b y_{n-1}'(t)y_{n-1}(t). n=1,2,3,...
There is no exact form of solution for the homogeneous part of your equation.

There is also the possibility to represent y(t) by a Fourier series
sum(Y[m]*exp(I*m*w*t),m=-infinity..infinity). (I^2=-1) Then, one has to solve for the coefficients Y[m] which means solving an infinite system of linear equations that is obtained by equating all coefficients of a particular exp(I*n*w*t) on both sides of the equation.

Furthermore, one may try Adomian decomposition and similar methods.

What the initial or boundary conditions for the differential equation? Are there conditions on the signs and relative sizes of a,b,c and A?

Due to the nonlinearity, I guess that the Homotopy Analysis Method (See the book of Shijun Liao "Homotopy Analysis Method in Nonlinear Differential Equations ") and/or the Homotopy Perturbation Method might be useful.

The whole difficulty lies in the nonlinear term. If b were zero this equation would have solutions that were linear combinations of exp(iwt) and exp(-iwt). (The simplest route to these solutions is to replace sin(wt) by -i exp(wt) and take the real part of the resulting solutions.) If b were small, perturbation theory could be used to get small b corrections to the b equals zero solution.

## All Answers (8)

Domingo López-Rodríguez· Brain DynamicsFrom this formulation it is quite difficult to obtain a closed-form solution for that ODE. I'd try a change z = some function of y, but I don't expect a solution to appear easily.

Instead, I'd suggest a numerical approach if the context allows you to do so. Finite differences methods are a good point to start with.

Best regards,

Domingo

Gro Hovhannisyan· Kent State UniversityThere is no exact form of solution for the homogeneous part of your equation.

Herbert H H Homeier· Universität Regensburgsum(Y[m]*exp(I*m*w*t),m=-infinity..infinity). (I^2=-1) Then, one has to solve for the coefficients Y[m] which means solving an infinite system of linear equations that is obtained by equating all coefficients of a particular exp(I*n*w*t) on both sides of the equation.

Furthermore, one may try Adomian decomposition and similar methods.

What the initial or boundary conditions for the differential equation? Are there conditions on the signs and relative sizes of a,b,c and A?

Mohamed E. Fouda· Cairo UniversityThis problem has zero initial conditions and the coefficients are positive without any conditions on on the relative size.

Herbert H H Homeier· Universität RegensburgR. Mark Bradley· Colorado State UniversityR. Mark Bradley· Colorado State UniversityArtur Sergyeyev· Silesian University in OpavaCan you help by adding an answer?