While introducing myself to differential equations, I read that the solution to a differential equation may contain an arbitrary constant without being a general solution.
I have been solving initial value problems under the concept of anti-differentiation for a long time now. I am a little aware of general and particular solutions from my engineering classes. I have always thought that a solution having an arbitrary constant cannot be anything other than a general solution. I don't think a particular solution can have an arbitrary constant in it. The lines that I read today left me curious. I am really curious to see these equations and maybe know how they work.
Best Answer
One can find examples (not exclusively) when the solution includes multi-valuated functions, which is the sometimes the case of inverse functions. For example, the ODE : $$\cos(y(x))\frac{dy}{dx}=1$$ has this family of solutions : $$y(x)=\sin^{-1}(x+c_1)$$ This solution includes an arbitrary constant $c_1$ but is not the general solution which is : $$y(x)=\sin^{-1}(x+c_1)+2\pi\:n$$