Differential Equation Solution

[EngWiPy]

Hi,

Does it necessary for a differential equation to have an analytic solution?

Regards

 

[HallsofIvy]

That depends upon what you mean by "analytic solution". If you have a differential equation that gives some property of "dy/dx", then you can certainly expect y to be differentiable. And, if you have a differential equation that says dy/dx= f(x,y), where f is a differentiable function of x and y, then it follows that
$$\frac{d^2y}{dx^2}= \frac{\partial f}{\partial x}+ \frac{\partial f}{\partial y}\frac{dy}{dx}$$
exists and then, by induction, all derivatives of y exist. Whether y must be "analytic" (if that is what you mean) is a little more complicated.