9.18.2 Greatest hits from previous weeks

9.18.2.1 How to find integer solutions?
9.18.2.2 1 Plot, 2 Scale/Axis
9.18.2.3 What is the equation to plot a vertical line?
9.18.2.4 How can I create an empty JSON object?
9.18.2.5 Find eigen energies of time-independent Schrödinger equation
9.18.2.6 How can I save both the plot and the legend together?
9.18.2.7 How to graph a sphere with cartesian equations?
9.18.2.1 How to find integer solutions?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=42359

Following is the equation: 
Need to solve for integer values of x,y when a,b lies between -1000 and 1000. 
Problem is to find the number of integer values of pair (x,y) that satisfy it. 
[equation-solving]
 
asked by Sahil Sareen https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=12396 Score of 2
answered by bill s https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=1783 Score of 4
9.18.2.2 1 Plot, 2 Scale/Axis

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=627

I would like to plot those two datasets on top of each other. But they have very different range on the $y$ axis. How can I have two different axis? 
 
I found the following on the help menu but quite ... 
[plotting] [customization]
 
asked by 500 https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=172 Score of 158
answered by ArgentoSapiens https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=92 Score of 160
9.18.2.3 What is the equation to plot a vertical line?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=59476

Please help. Trying to find area between three curves,  e^-x, x = 2, y = 1 . Can't find out how to plot  x = 2 . Don't want to use <... 
[plotting] [calculus-and-analysis]
 
asked by Cory https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=19717 Score of 7
answered by Basheer Algohi https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=13548 Score of 13
9.18.2.4 How can I create an empty JSON object?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=50727

For creating non-empty object, I use following syntax 
 
 ExportString[{"key" -> "value"}, "JSON"] 
(*{"key" : "value"}*) 
 
 
But how can I create an empty object? ... 
[export] [conversion] [json-format]
 
asked by molekyla777 https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=11981 Score of 10
answered by jkuczm https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=14303 Score of 11
9.18.2.5 Find eigen energies of time-independent Schrödinger equation

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=32293

I'm trying to get the eigenvalues of a one dimensional time-independent Schrödinger equation, 
 
$-\frac{h^2}{2m_0}\frac{d^2\psi}{dx^2}+U(x)~\psi=Ei~\psi$ 
 
where U(x) is some potential and Ei is the ... 
[equation-solving] [physics] [parametric-functions]
 
asked by xslittlegrass https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=1364 Score of 51
answered by Jens https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=245 Score of 47
9.18.2.6 How can I save both the plot and the legend together?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=35653

In this example, how can I put the legend inside the graph? Currently, when I save the graph, only the graph is saved, not the legend. 
 
... 
[plotting] [legending]
 
asked by Mika Ike https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=6543 Score of 45
answered by C. E. https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=731 Score of 45
9.18.2.7 How to graph a sphere with cartesian equations?

https://mathematica.stackexchange.com/landing/r/digest?cta=question&id=24169

I'm new to Mathematica and I need to ask how to graph a sphere, cylinder, etc.. using cartesian equations. 
 
For example in sphere's case $x^2+y^2+z^2=1$ 
 
I tried to use ... 
[plotting]
 
asked by German https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=7145 Score of 9
answered by Dr. belisarius https://mathematica.stackexchange.com/landing/r/digest?cta=user&id=193 Score of 15