Schrodinger's Equation and Wave Function
I suspect the problem is not mathematical. It is in the meaning of the wave function. Well. I will now show you how I explain it qualitatively in my high school classes. I state that the electron is like a violin string, but a string with three dimensions that is vibrating in the fourth dimension. As I know this last words are not understandable, I start with one dimension and pass slowly to two and three dimensions.
Let's start from a horizontal string whose end is fixed on a wall, and the other end is in your hand. If you put some energy to the string, it will vibrate in a stationary vibration. The string looks like half a sine curve, with the maximum in the middle of the curve. If you want to describe the curve with words and without drawings, you can say that the curve is like a half-sinusoid whose amplitude has the same sign all over its length, part time positive, part time negative.
Now if you put more energy in the string, it will vibrate in another mode, with a node in the middle. When one half of the string is vibrating upwards, the other half is vibrating downwards. With still more energy you can have it vibrate with two nodes. To describe the shape of the string as before without drawings, you can say that the string is a sinusoid with one or several nodes. So that you need one number $n$, the number of nodes, to describe its movement : $n$ is the first quantum number. $n = 1$ for the first vibration, because the only point not moving is the end of the string in the wall. $n = 2$ for the case with one node in the middle of the string.
Now let's consider the string in two dimensions. This is not a string any more, but a two-dimensional plate like a drum. If you hit it softly in the middle, it will vibrate with all its points moving simultaneously upwards then downwards. The only points of no vibrations are the periphery of the drum. In two dimensions, points of no vibration are replaced by lines of no vibration. Here there is only one such line : the number of nodal lines is $n = 1$.
If you put more energy by hitting the center of the plate, you may produce a new vibration mode if the center of the plate is moving upwards (or downwards) when the outer part is moving in the opposite direction. In this mode no vibration occurs on the small circle between these. This circle is a nodal line. There are two concentric circles on this mode. So  $n = 2$. We may describe its amplitude by drawing curves of equal amplitude on the plate, like level curves on a map.
But it is going to be more difficult, because there is another type of vibration on the plate. If you hit the plate not in the center, but elsewhere, you may produce a plate vibration where one half of the plate moves upwards and one half downwards. Here the nodal line is a diameter. And there are two nodal lines : the diameter and the periphery. Here $n = 2$. As a consequence, two vibration modes exist with the same quantum number. We must introduce a second quantum number, which is called $l$, and which is the number of straight nodal lines. So this type of vibration is given by $n = 2$ and $l = 1$. Hopefully you understand that other vibration modes do exist with the drum, with arbitrary values of $n$ and $l$ &lt; $n$
Let's summarize. An object with one dimension vibrates in another dimension, and one quantum number is enough to describe its vibration. An object with two dimensions is vibrating in the third dimension, and two quantum numbers are required to describe its vibration. The number of dimension is equal to the number of quantum numbers.
Now let's go to the electron, which has $3$ dimensions, and which may be considered as if vibrating in a fourth dimension, called $\Psi$. As this dimension cannot be represented in our world, all we can do is to give different numerical values to three quantum numbers for all vibrating modes, and represent surfaces which correspond to equal numerical values of the local &quot;vibration&quot; of the electron. These surfaces of equal value of $\Psi$ are spheres for $l = 0$, the famous dumb-bells with two lobes for $l = 1$, because of $1$ nodal plane. Here is how it goes.
Here is how I explain the wave function to my high school students. They usually have some difficulty understanding. But all have told me that this explanation had helped them sudying quantum mechanics later on in college.