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 is instantaneously mixed thoroughly and flows out of the cooler at a cons tant rate of 3 litres per minute through another outlet. (i) If x grams is the amount of chemical X in the cooler t minutes after the coolant is added, show that x satisfies the differential equation d3 120d5 0 x x tt  . [2] (ii) Find the amount of chemical X in the cooler when there are 51 litres of mixture in it. [6] The temperature of the mixture, 𝜃 oC follows the differential equation  0.04d 60e cos(0.01 ) 1d t t   . Use the improved Euler method with step size 0.5 to estimate to 1 decimal place, the temperature of the mixture 1 minute after the start of the addition of coolant. [4] [VJC/H2 FM/2017/Prelim/P1/Q11] S/N Answers 1 (ii) 22yxA x (iii) 22 (1 )1yx  2 tan secyt c t (i) tanyt (ii) tan 2secyt t 3 (a) tan 21exc   (b)  2 1 ln y Cx x x   4 (a)(i) 1 e xyB   (ii) 1 2e xy   (b)(i) 1 2058 0 e t Vt C   5 (a) 2 exyx C x (b) 0.9225 (4 d.p.) 6 (i) f ( 0.2) 1.241 (ii) 2 3 2 d 24 2d y xy yx   7 (ii) 117 grams (3 s.f.) ; 50.9 oC (1 d.p.).