How can I intuitively predict the shape of a solution to ordinary differential equations? I am new to differential equations, and I realized that I don't have even an intuition as to what solutions to ordinary differential equations would roughly look like. For example, given the governing equation:
$$ \frac{\partial^2 y}{\partial x^2} -\frac{\partial y}{\partial x} + y = 0 $$
for the range $$0<x<10$$
given the initial conditions $$y(0) = 1$$
$$y(10) = 5$$
The plotted graph for the function y looks like:

How can you intuitively predict that y will be mostly flat in the range 0<x<6?
If so, how would you intuitively picture the solution for this function y without solving for it analytically?
 A: With a little computation one sees that the characteristic polynomial
$$
0=r^2-r+1=(r-\tfrac12)^2+\tfrac34
$$
has a complex pair of roots that result in an oscillatory solution with an amplitude growing like $e^{x/2}$. So an amplitude of about $1$ at $x=0$ grows to $e^5=148.4...$ The small value at $x=10$ can thus only be realized by locating a node of the oscillation close-by. As the wave length of the oscillatory factor is $7.255...$ the graph will thus contain one-and-a-bit periods, thus with a maximum of the amplitude at about $x=7$ to $8$.
A: I don't think that it is possible to predict the shape of a general ordinary differential equation (ODE),i have found this tool useful to look at the behaviour of ODE's, it is best for first order ODE's.
but there are some that are more common (a quick note $A$ and $B$ are arbitrary constants)

*

*for example the simple harmonic oscillator
$$ \frac{\partial^2 y}{\partial x^2} +k^2 y = 0 $$
which has solutions $y=A \sin(kx)+B \cos(kx)$


*a related ODE is this one $$ \frac{\partial^2 y}{\partial x^2} -k^2 y = 0 $$
which has solutions $y=A \sinh(kx)+B \cosh(kx)$


*then there is the damped simple harmonic oscillator
$$ \frac{\partial^2 y}{\partial x^2} +k_0\frac{\partial y}{\partial x} +k_1^2 y = 0 $$
which has solutions $y=e^{x\alpha}(A \sinh(x\omega)+B \cosh(x\omega))$ for constants $\omega$, $\alpha$ which depend on $k_0 $ & $ k_1$.
There are also solutions of commonly occurring ODEs that have less pretty forms

*

*Bessel functions, used in determining waves on a circular plate.

*Legendre polynomials, used in the describing spherical waves in 3d, and used to describe atomic orbitals.

*Chebyshev polynomials, used in signal analysis

*Hermite polynomials, several uses in signal analysis, and in the description of some quantum states.

Hopefully that helps
A: Slope Fields offer a good visual aid, sometimes almost showing the actual solutions. You only calculate the slope at select points using the given formula for the first derivative, so it's usually a somewhat less intensive process than solving numerically.

Phase Portraits Have similar features and are more generalizable to second order differential equations.

