How can you change DPI in Photoshop without changing the size?

This a complicated question :)

when you change DPI you only change number of pixels per Centimeter or Inch, but absolutely you change the display size of image over your screen, this is a different factor of scaling.

for screens actual pixels resolution is72 pixels per inch, if you resize more or less your display will map you image to fit the same size. obviously you can virtually zoom in and zoom out but there is a fitting size for your actual image size.

So when you change DPI you don not change printing size but you force the change of display size.

Load your image into Adobe Photoshop (or Photoshop Elements).

2.            Select the "Image > Image Size" menu option (may be "Image > Resize > Image Size" in Photoshop Elements).

3.            In the image size dialog window, deselect the "Resample Image" checkbox (make sure there is no checkmark in that box).

4.            In the "Resolution" box type in300 (or whatever DPI you want)

5.            Click the "OK" button

6.            Your image DPI has now been set to whatever you want (leaving the pixel dimensions of the image unchanged).

7.            Save this photo with a new name - I suggest adding a -300dpi extender (i.e. "dpi.jpg") to identify this new DPI image.

8.            See note below when saving to JPEG format

go to image size option, turn off the constraint proportion and change dpi

Image> Image Size> Set Re sample Unchecked, Change Resolution to desired DPI.

But Increasing Resolution may not help in actual pixel increasing, Depending on the actual image quality, depth of colors and pixels. but reducing DPI helpful for decreasing file size for digital print or for online publishing.

select image > press ctrl+shift+I >resolution

changing DPI is different than changing the Project Scale over all, with increasing or decreasing the number of the pixels per inch\\centimeter we will have more effecint image -from the menu bar:-IMage->Image Size-at the Image size window turn off the "Resample" option -then change the DPI as you want.

You can change in image size at image menu. change the resolution.

1. Begin with the file you created in the previous set of steps, and return to its original state using the History panel. If you closed it, then go back and complete Steps 1 and 2.Tip: If you still have the Image Size window open from the preceding example, you can reset it to the settings it had when you opened it by holding down Alt and clicking Reset—where the Cancel button used to be. This works with almost every dialog box where you see a Cancel button.
2. Choose Image > Image Size (Alt+Command+I/ Alt+Ctrl+I). The Image Size dialog box appears.
3. Leave the Resample Image option selected or select it if it isn’t already checked.
4. Change the Resolution to 300. Leave the dimensions at pixels/inch. Notice that the Image Size remains at 8" × 10". What changed was the Pixel Dimensions at the top of the window. They went from 1200 × 1500 to 2400 × 3000.
5. Change the Resolution to 600. Now the Pixel Dimensions changes to 4800 × 6000, but the Document Size remains fixed, as shown below. 

you can click on the littel gear-look in the upper right of the "image size" dialogue box, you will find "scale styles" option, it allows you to change resolution without change the size of the image 

By checking the 'resample' button in the image size window.

By changing the resolution on Image Size (cmd+alt+i)