As has been posted earlier mastering differential equations is an exercise in symbol manipulation, but the underlying equations are really important.
Mathematics is an ordering of nature via symbols. In the ordering of nature, Newton realized that most equations had a second level of ordering that described the original equation. These equations of differentiation and integration were achieved by making differencing ratios and approaching a limit. Differential formula can be used in every field of science. They are used regularly in Computer Science usually as an algorithm to optimize a process.
Learning to manipulate these equations in your situation is probably unnecessary. Understanding what these equations are used for in the real world is very useful. I suggest you consult Google for each equations use in real world situations, if only to give you some mnemonic for learning this stuff. (You can probably consult Google for the DE problem answers too.) If you know how the equation/formula is used in the real world you might see a use for the same concept in a program, hence it is good for a degree in CS.
On the flip side, a good teacher should be able to make this stuff come alive and be far less dry then you make it out to be. Your academic career will flourish if you spend a lot more time researching your teachers for next semester. Consider a different institution if the student consensus is that there are no good math teachers where you toil.