Solve the differential equations dx/dt=2x+y+1 and dy/dt=4x-y+1 given that when t=0 x=20 and y=60. (A2 Further pure)
These questions are really mean but if given in the exam give ALOT of marks and whilst scary at first follow a very general pattern. First I would take either equation and re-arrange for the lone variable. Take the first equation:dx/dt=2x+y+1 re-arrange for y y=dx/dt-2x-1then differentiate with respect to x dy/dt=d2x/dt2-2dx/dtthen realise we have two equations that begin with dy/dt so equate them with eachother to obtain d2x/dt2-2dx/dt=4x-dx/dt+2x+1+1here I would point out to substitute for the lone 'y' in the equation and watch out for the negative sign by the y as it swaps the sign of all three functions in the 'y' equation. I would then re-arrange this formula to be in a suitable form which after manipulation would be :d2x/dt2 -dx/dt-6x=2. Then I would do the bog standard solving as it is straight forward from this point creating an equation of m2-m-6=0 for the complementrary function, and since this is a simple quadratic acknowledge that the particular integral will simply be in the form x=Z. Then after differentiating x with respect to t, the two proceeding differentials will be o, and then submitting x=Z into the equation formed would give -6Z=2. Then to solve for Z you obtain -1/3. Then finally after all of this you have the first equation for this question to deal with. You would then do the exact same for the other equation, end up with two equations and then solve by simply putting in the numbers. This question is one of the more difficult that appear in the further maths spec and require alot of set up, but with practise are very do-able.
