Here you can find several example questions from STEP past papers for you to practice your skills on. There's questions covering formulating your own differential equation, as well as solving first order and second order problems: everything you need to get started. There's even hints to help you along if you need them.
Question 7 STEP I 2011
Hint: What form must the differential equation take for the derivative to be stationary at the required height? How do you solve a differential equation of this type? Can you solve for \(T\) at the height \(\alpha H\)?
Hint: What does our differential equation change to? What substitution will let you integrate this separable differential equation? Can you follow the same method again to find \(T'\) for the required height?
Question 7 STEP II 2008
Hint: Can you implicitly differentiate \(y\) to substitute in for \(y\) and \(dy/dx\) in the differential equation? What type of first order differential equation are you left with? How do you solve these?
Hint: How can you adapt the substitution method from the first part to the differential equation here?
Hint: What pattern can you spot from the solutions to the two previous problems? How would this generalise to the case here?
Question 6 STEP I 2010
Hint: Can you differentiate \(y\) twice to substitute into \((*)\)? What about for \(y=ue^x\)?
Hint: If you substitute in for \(v\) what type of first order differential equation do you have? Can you solve back then for \(y\)?