How can I repair a clog in the drain by my washing machine?

Get your self a metel hanger and make a U shaped bracket that will hold the hose inside the sink and attach on the other part of the hose to stay in a U shape to hook on to side of sink

Duck tape!!

Here are perfect instructions (I LOVE SF gate!)

https://homeguides.sfgate.com/keep-washing-machine-hose-falling-floor-20836.html

Scroll down to page 3. There are minimum hose diameter requirements, etc. It sounds like in your case you need a U-shaped hose support.

If quite old, the standpipe may not be large enough to handle the capacity of the washer's rinse load. "A stand-alone connection to a washing machine drain hose is the most desirable option. This means the drain pipe is indeed designed to handle the kind of workload or volume of water typically drained from a washing machine. A dedicated drain pipe is usually a larger diameter than a typical sink drain. It is less likely to clog, or dislodge. It usually also connects at a higher elevation, well above the machine itself, another advantage."

http://www.balkandraincleaning.com/washing-machine-drain-hose-connections-maintenance/

If you are VERY lucky, the problem is just a connection as is, or extending the standpipe up higher.

2 diagrams of using zip-ties or a cable tie

https://www.howtolookatahouse.com/Blog/Entries/2018/6/is-a-washing-machine-drain-hose-required-to-be-secured-at-the-standpipe-1.html

Pipe clamps you screw tight..... https://www.homedepot.com/p/Seachoice-Hose-Clamp-Set-in-Stainless-Steel-23501/306288018?cm_mmc=Shopping%7CG%7CBase%7CD25H%7C25-3_FASTENERS%7CNA%7CGeneric%7CPLA%7c71700000052227365%7c58700005026398223%7c92700046753314620&gclid=CjwKCAjwzJjrBRBvEiwA867bynlpri0wsvfibTew12kBAG34KmQGYpyVPn99LydaQugKI_orSyh6BBoCC14QAvD_BwE&gclsrc=aw.ds any hardware store sells these individually, just need to know diameter of the pipes.

You might also run a drain snake down that pipe to clean out the lint buildup.