If your ODE is of the following form
$$x^{(n)}(t)=f(t,x^{(0)}(t),\ldots, x^{(n-1)}(t))$$
and $f:\mathbb{R}\times\mathbb{R}^n\rightarrow\mathbb{R}$ is locally Lipschitz, then the general solution has to include $n$ arbitrary constants. It boils down to an application of the Picard-Lindelöf theorem. You can transform your ODE into the following form:
$$y'(t)=F(t,y(t))$$
with $F:\mathbb{R}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^n$ locally Lipschitz. Then you can impose an arbitrary initial condition $y_0=y(\hat{t})\in\mathbb{R}^n$ and Picard-Lindelöf gives you a unique solution. Therefore you can impose exactly $n$ arbitrary constants, which have to be reflected in the general solution.
A counterexample can be found by looking at nonunique solutions of ODE's, for example $x'=\sqrt{x}$ with $x(0)=0$. it has at least two solutions, $x_1(t)=0$ and $x_2(t)=\frac{t^2}{4}$ but you can combine these two, to obtain a more general set of solutions like this:
$$x(t)=\left\{\begin{array}{cc}\frac{(t-t_0)^2}{4},& t\geq t_0\\ 0,& t<t_0\end{array}\right.$$
with $t_0\geq 0$. Hence you can impose an initial condition and at least another parameter $t_0$.
To the PDE Question of things: For PDE's there is no general uniqueness statement, therefore you cannot expect to find a general solution which allows you to prescribe a given number of parameters. Even if you have uniqueness the number of prescribable parameters usually becomes infinite. Let me explain what I mean in an example: Imagine you have the following PDE $\Delta u= f$ on a compact smooth domain $\Omega\subset \mathbb{R}^n$, with a given $f:\Omega\rightarrow\mathbb{R}$ at least integrable. Then you can prescribe so called boundary conditions, which will make the solution of the problem unique. What I mean is the following: You can prescribe $u|_{\partial\Omega}=g$ with $g:\partial\Omega\rightarrow\mathbb{R}$ for example continuous. Since $\partial\Omega$ has infinitely many points, you have prescribed infinitely many parameters.
As an additional comment: You might want to check out Green's functions, they give you a kind of general solution to some linear PDEs. Here is a wiki link: https://en.wikipedia.org/wiki/Green%27s_function
