# Modern methods for ODE with complex harmonic oscillations and phasors

We have a differential equation of the following form:

$$\frac{dx}{dt}=f(x) \cdot a \cdot \sin(\omega \cdot t)-a \cdot \sin(\omega \cdot t + \frac{\pi}{2})$$

where $$f(x)$$ - arbitrary function from state variable, $$a$$ and $$\omega$$ - amplitude and frequency of harmonic.

The right side of this equation can be determined by harmonic oscillations addition, and the equation itself can be written in the following form:

$$\frac{dx}{dt} =\sqrt{a^2+a^2 \cdot f(x)^2} \cdot \sin(\omega \cdot t + \phi(x))$$ (1)

where $$\phi(x) = \arcsin(\frac{a^2 \cdot f(x)^2}{\sqrt{a^2+a^2 \cdot f(x)^2}})$$ - auxiliary phase

List of trigonometric identities - Linear Combinations

In addition, complex harmonic oscillation can be considered as a summation of phasors. Then, the differential equation takes the following form:

$$\frac{dx}{dt} = f(x) \cdot a \cdot \cos(\omega \cdot t - \frac{\pi}{2})-a \cdot \cos(\omega \cdot t) = \mathrm{Re}(f(x) \cdot a e^{-i \cdot \frac{\pi}{2}} e^{i \cdot \omega \cdot t}) + \mathrm{Re}(-a e^{i \cdot 0} e^{i \cdot \omega \cdot t})$$ (2)

DSolve[x'[t] == x[t]^4 Sin[t] - Cos[t], x[t], t]

During evaluation of In:= Solve::ifun: Inverse functions are being used by Solve, so some solutions may not be found; use Reduce for complete solution information.

Out= DSolve[Derivative[x][t] == -Cos[t] + Sin[t] x[t]^4, x[t], t]


This equation is non-linear, and it’s unlikely that it can be solved “head-on” using DSolve. It is interesting to try to solve it using coordinate transformation, Lie symmetry groups, or the method of homotopy analysis.

1. Are these tools available in Mathematica?

2. Is it possible to obtain approximate solutions for qualitative and quantitative estimates of transient parameters. Expansion in rows is unacceptable, because due to the complexity of the solution, a large number of members of the series are required, and cutting off the excess makes it necessary to lose important information about the properties of the system.

• this is Chini ODE. Very few analytical solutions exist for this type of ode. Jun 22, 2020 at 5:03
• DSolve[x'[t] == x[t]^4 Sin[t] - Cos[t], x[t], t] - Are you talking about this equation? $\frac{dx}{dt}=f(x) \cdot a \cdot sin(\omega \cdot t)-a \cdot sin(\omega \cdot t + \frac{\pi}{2})$ - In general, I'm interested in ways to solve this equation. Perhaps this is an even more difficult task?
– dtn
Jun 22, 2020 at 5:07
• I am talking about this ODE x'[t] == x[t]^4 Sin[t] - Cos[t] which you had in your code. There is no analytical solution for it. At least Mathematica and Maple think so. Jun 22, 2020 at 5:09
• Yes, solving non-linear equations is a difficult task and a big mathematical problem ... Is it possible to somehow evaluate the qualitative and quantitative characteristics of such systems at least approximately?
– dtn
Jun 22, 2020 at 5:14

Is it possible to somehow evaluate the qualitative and quantitative characteristics of such systems at least approximately?

One possible way is to use AsymptoticDSolveValue and obtain series solution approximation around different t values. For example, at t=0

AsymptoticDSolveValue[x'[t] == x[t]^4 Sin[t] - Cos[t], x[t], {t, 0, 4}]


gives

$$\frac{1}{24} c_1{}^2 \left(36+12 c_1{}^5-c_1{}^2\right) t^4+\frac{1}{6} \left(1-8 c_1{}^3\right) t^3+\frac{c_1{}^4 t^2}{2}-t+c_1$$

This is a Chini ode. Only special cases have known analytical solutions.

ode:=diff(x(t),t)=x(t)^4*sin(t)-cos(t);

[_Chini]


Maple can not solve it either. You could ask at math group, may be someone can come up with way to solve it.

• Have you met the Homotopy Analysis Method (HAM)?
– dtn
Jun 22, 2020 at 5:33
• And one more question. Can I use the Fourier series in AsymptoticDSolveValue?
– dtn
Jun 22, 2020 at 5:36
• about HAM, saw it mentioned before, but never studied it nor used it. I am not sure I understand you question about using Fourier series in AsymptoticDSolveValue? The series solution given in AsymptoticDSolveValue is not a Fourier series, if this is your question. Fourier series is global approximation of a function, while series given by AsymptoticDSolveValue is local approximation, valid close to point of expansion only, and not valid everwhere like with Fourierseries. Jun 22, 2020 at 5:39
• sciencedirect.com/science/article/pii/S089396591000193X
– dtn
Jun 22, 2020 at 5:40
• about Fourier Series, Yes, I understand the differences between these series. The fact is that a numerical analysis shows that the solutions of this equation contain a strong periodic component. In this situation, it is best to use the Fourier series at a given time interval.
– dtn
Jun 22, 2020 at 5:42