About Solvable/Unsolvable ODEs

[john.lee]

In my class, I learned about a First-order ODEs,

and solvable and unsolvable.

example in case solvable ODEs)
dy/dt=t/y
dy/dt=y-t^2

example in case unsolvable ODEs)
dy/dt=t-y^2

but , i don't know how distinguish those.

please, teach ME! : ( as possible as easily !

 

[hilbert2]

The question is not about whether the equation has a solution. It's about how easy it is to find the solution. The equation $\frac{dy}{dt}=\frac{t}{y}$ is separable and is easy to solve by multiplying both sides with $ydt$ and integrating the resulting equation $ydy=tdt$. The equation $\frac{dy}{dt}=y-t^{2}$ can be solved by first multiplying both sides with the integrating factor $e^{-t}$, and then using the derivative of product rule and integrating the resulting equation $\frac{d}{dt}\left(e^{-t}y\right)=t^{2}e^{-t}$.

For the nonlinear and non-separable equation $\frac{dy}{dt}=t-y^{2}$, there is no similar simple method of solution. There does exist a solution, but it must be written in terms of special functions called Bessel functions. Do you know how to solve DE:s with WolframAlpha or Mathematica?

 

[john.lee]

Oh,! I got it! Thanks : )