What causes students not to be interested in problem solving? Sometimes students lack the drive for problem solving. One might say that they are bored.

However, is there a scientific account on what causes ignorance of problem solving, when they simply are not interested in solving any problem? Like when they think that it is the grown-up's task to solve problems, not theirs.
Nature supplies each of us with a Homo Sapien type brain with 100 billion neurons and 100 trillion synapses. However the brain is plastic and can change its structure based on the way it is programmed psychologically. 

Not all people are capable of solving all problems. Our abilities mental and physical are based on what we spend our time doing. Even the banal task that one could imagine is quite exhilarating to another person with a brain which was trained and programmed differently. 

First the brains structure is made up of the connections the neurons make. My favorite study which shows this is the Cab Drivers Brain Experiment. It showed that after years of labor (and compared to a control group) London taxi drivers had enlarged in structure posterior hippocampus giving the cabbies the ability to store large amount of navigational data in their brain. (That is not to say that the cabbies all enjoyed driving though.) Another example of this includes changes to accommodate musical ability. Exploiting neural plasticity has led to treatments for dyslexia. Likewise also everyone who uses the internet has seen ads for Luminosity but not everyone knows that it doesn't work well. 

People who are mentally ill have connections in their brain that lead them to illness. These connections can be caused by physical or by self-psychological training. That is to say chemical imbalances are often the result of a structural or physical malformations. A brain may for whatever reason produces too much or little neurotransmitter or have many connections to an incorrect part of the brain. If the brain was a terrane the paths traveled more often would eventually become "paved" (somewhat permanent). It take medication and counseling to break up those bad pathways and form new and healthy pathways.

Nature or nurture has resulted in the group of people which I think you describe as somewhat dependent or ADHD. Only proper counselling and perhaps even some medication will be able to free them from their lack of motivation. Even with proper treatment some people will for whatever reason never want to act in the way you see fit. Homework will always be at times left undone. This is a truth no matter how advanced the teacher is in academia or how driven and motivated the pupil.