DHS First-order Differential Equations (9649) Topical Revision

Differential Equations I H2 Double Math Differential Equations I [part FM] 1 (i) By means of the substitution ,yv x show that the differential equation 22d2, 0 d yxy y x xx   can be reduced to 2d1 .d2 vv xv x  [3] (ii) Hence find the general solution. [4] (iii) Sketch the solution curve passing through (2,0). [2] [SRJC/Promo/2016/9] 2 Find the general solution of the differential equation dcos sin cos ,d yty t tt  where 11 22ππ ,t giving your answer in the form f( ) .yt [3] On a single diagram, sketch (i) the solution curve that passes through the origin, [1] (ii) the solution curve which has a minimum point when 1 6 π.t [4] [AJC/Prelim/2005/I/15(b)OR] 3 (a) Find the general solution of the differential equation  d1c o s 1 d x x   . [5] (b) Show that the differential equation 2d 2l nd yy yxxx may be reduced by the substitution 1y u to d 2l nd uu xxx  . Hence find the general solution for y in terms of x. [5] [MJC/Prelim/2005/I/11]

Differential Equations I 4 (a) (i) Given that 0x , find the general solution of the differential equation 2 d ,d yxy x  expressing y as a function of x. [3] (ii) Sketch the graph of the particular solution for which 2 as .yx  [3] (b) Water starts pouring into an empty open tank, and t seconds later the volume, V litres, of water in the tank is given by d1 1 1.d2 0 4 V Vtt   (i) Find V in terms of t, and hence show that, for large values of t, 58 0 .Vt [5] (ii) What can we deduce from (b)(i), in the context of this question? [1] 5 (a) Find the general solution of the differential equation  32d1, d yx yx x x x  leaving your answer in the form f.yx [5] Sketch and label clearly the equations of 2 distinct members of the family of solution curves where their nature of stationary points differ from each other. [2] (b) It is given that  2d 3 and 0 1.d N tN Nt   Use two iterations of the improved Euler’s method to find an approximate value of  0.4N , correct to 4 decimal places. [4] [NJC/H2 FM/2017/Prelim/P2/Q4] 6 It is given that f( )yx is a particular solution of the differential equation 2d 2d y x yx  with the initial condition f( 0 ) 1 . (i) Use the Euler method with two step s of equal size, starting at 0x  , to approximate f ( 0.2) . [2] (ii) Find 2 2 d d y x in terms of x and y. [1] Determine whether the approximation found in (i) is an under-estimate or an over-estimate. [2] [RI/H2 FM/2017/Prelim/P1/Q5b]

Differential Equations I 7 An industrial cooler initially contains 50 litres of pure water. When the temperature of the water rises to 60 oC, a coolant containing 30 grams of chemical X in every litre of water is added into the cooler at a constant rate of 4 litres pe r minute. The mixture i