RI Differential Equations Solns

RAFFLES INSTITUTION H2 Mathematics 9758 2023 Year 6 Term 3 Revision 12 (Summary and Tutorial) Topic: Differential Equations Summary for Differential Equations Differential equations of the form d f( )d y xx = - solved by “direct” integration, i.e. integrate f( )x with respect to x. d f( ) f( ) dd y x y xxx = ⇒= ∫ Example [9740/2008/01/Q4(i)(ii))] (i) Find the general solution of the differential equation 2 d3 d1 yx xx= + . (ii) Find the particular solution of the differential equation for which 2y= when 0x= . Solution: (i) 2 d3 d1 yx xx= + 22 2 2 33 d [integrate the f( ) with respect to ]11 32 d21 3ln( 1)2 xxyx x xxx x xx xc = =++ = + = ++ ∫ ∫ (ii) When 0, 2xy= = 232 ln(0 1)2 c= ++ 2c= Thus the particular solution is 23ln( 1) 22yx= ++ .

Differential equations of the form d f( )d y yx = - solved by separating the variables, i.e. d1 f ( ) d 1 dd f( ) y y yxxy= ⇒= ∫∫ Example [9740/2012/02/Q1(b)] Given that u and t are related by 2d 16 9d u ut = − and that 1u= when 0t = , find t in terms of u, simplifying your answer. Solution: 2 2 22 d 16 9d 11 d 1 d [ separate the variables to obtain d 1 d ]16 9 f ( ) 13 d 1 d3 4 (3 ) 1 1 43 ln3 2(4) 4 3 1 43ln24 4 3 u ut ut utuu utu u tcu u tcu = − = =− =− +×= +− + = +− ∫ ∫ ∫∫ ∫∫ When 0, 1tu= = 1 ln724c= Thus 1 43 1 1 43ln ln 7 ln24 4 3 24 24 7(4 3 ) uut uu ++= −= −−

Differential equations of the form 2 2 d f( )d y xx = - solved by integrating the right hand side twice with respect to x. Note that 2 2 dd f( ) f( ) ddd yy x xxxx = ⇒= ∫ Now suppose that f () d F ()xx xc= +∫ , then we have d F( )d y xcx = + which gives F( ) dy x cx= +∫ Example [9740/2012/02/Q1(a)] Find the general solution of the differential equation 2 2 2 d 16 9d y xx = − Giving your answer in the form f( )yx= . Solution: 2 2 2 2 3 3 24 d 16 9d d 16 9 d [integrate with respect to ]d 16 3 16 3 d [integrate with respect to ] 38 [note that there are two constants and ]4 y xx y xx xx xxc y x x cx x x x cx d c d = − = − =−+ = −+ = − ++ ∫ ∫

Solving Differential Equations using a given substitution - the substitution will be given in the question - this substitution allows us to reduce the differential equation to familiar forms which we discussed earlier, namely d f( )d y xx = or d f( )d y yx = . Example [9233/N87/02/Q14(a)] The variables x and y are connected by the differential equation d1 d1 y xy x xy ++= −− . Show that the substitution uxy= + reduces the equation to d2 d1 u xu= − , and solve this differential equation. Solution: dd 1 [ differentiating both sides with respect to ]dd uyuxy xxx=+⇒ =+ dd 1dd yu xx∴=− d1 d1 d1 1d1 d1 1d1 11 1 2 (shown)1 y xy x xy uu xu uu xu uu u u ++= −− +−= − += +− ++−= − = − 2 (1 ) d 2 d [separate the variables] 1 22 uu x u u xc −= −= + ∫∫

Differential Equations in Real-World Context - this includes formulating the differential equation in the given context, solvi