# How to solve this differential equation d^2 y/d^2 t +(a+by)dy/dt+cy=Asin(wt)?

This equation results from modeling a nonlinear element.

Question

This equation results from modeling a nonlinear element.

- Hello,

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.

Best regards,

Domingo - 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? - Thanks for your answer

This problem has zero initial conditions and the coefficients are positive without any conditions on on the relative size. - 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.
- A resonance occurs if a=0 and w^2 = -c. If a is nonzero but small, perturbation methods will fail for w close to the resonant frequency.
- For a=0 and c = w^2/4 I have found a particular (not general!) solution y(t)=Q sin(wt/2), where Q=2(A/(wb))^(1/2).

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