# How can I find explicit solutions of this ODE $\dot x=e^x\sin x$?

In order to solve this question: Interval of definition of the solutions of $\dot x=e^x\sin x$

I'm trying to solve this ODE $\dot x=e^x\sin x$ without success, can I have an explicit solution of this ODE?

Following the comments note that $x$ is defined in $\mathbb R$.

I need help

Thanks a lot

• separate and integrate when $x \neq n \pi$ of course the constant solutions $x = n\pi$ are exceptional in this problem. For the case of a vector $x$ I have no idea what $\sin(x)$ would mean. – James S. Cook Apr 8 '13 at 2:39
• Semi-echoing what JamesS.Cook said--how do you define $\sin(x)$ and $\exp(x)$ for vector $x$? For square matrix $x$, it's easy. For vector, not so much... – apnorton Apr 8 '13 at 2:44
• @JamesS.Cook so do you mean, this ODE makes sense only in R? – user42912 Apr 8 '13 at 2:52

## 1 Answer

As others have pointed out the function $$\frac{dx}{dt}=f(x)$$ can be explicitly integrated by $$\frac{dx}{f(x)}=dt\rightarrow \int \frac{1}{f}+C=t$$If $f(x)\not=0$ at that point. If $f(x)=0$ then $x=x$ represents a fixed point for the flow given by the differential equation. For our problem the integral for $\int \frac{1}{f}$ is quite ugly (see here) and it is questionable if you can find it of much use. I suggest you try to draw the integral curves by hand instead, which is not difficult.

• At his previous question on this topic I put an answer for $y' = \cos y,$ which is similar (phase portrait, anyway) to the question asked and has closed form solutions. Not sure he has looked at it. – Will Jagy Apr 8 '13 at 21:28