Please solve the attached questions (2) Use appropriate substitution to solve the following(i)dytan (x+y). (ii) y:1-1-y%3D%3Ddr(iii) y" + (y/)² + 1= 0. (iv) xy"+ (v/)² = 0.Question Four(1) In a third-order chemical reaction the number of grams X of compound obtained bycombining three chemicals is governed bydX= K(a - X)(8 – X)(-X).dtSolve the equation under the assumption a B #y.(2) The population P(t) at any time in Ras Al Khaimah is governed by the initial-valueproblemdP= 0.1P – 0.0004P?,dtP(0) = 200,where t measured in year. What is the limiting value of the population?. At what time willthe population be equal to one-half of this limiting valuc?.(3) In one model of the changing population P(t) of a community it is assumed thatdP dBdDdtdtdtwhere dB/dt and dD/dt are the birth and death rates; respectively.(i) Solve for P(t) ifdB= KP anddD= K2P.dt(ii) Analyse the cases K1 K2, K1 = K2, K1 0 constant )dy(i)dr- sin*(z)y = cos (z)y + 1. (ii) 7ty – e'y' =t2 + 4adr(iii) zª + (a² – 2a)z-- tela-2)=. (iv) y" = y² + cos(x).dt%3D(2) Find the values of r, if(i) / = 7x, y = rz². (i) y" – 7y" + 6y/ = 0, y = e".(iii) ty" + 8ty/ + 6y = 0, y = t".Question Two(1) Show that z = A cosh(wx)+ B sinh(wr) satisfies the differential equation z"-w²r = 0.(2) given that y(t) = ae' + bet with y(0) = 2 and y/(0) = 2, find the values of a and b.%3D(3) The general solution to second-order differential equation 5y" = 2y is in the formy(x) = ce" + C2, where c and cz are constants. Find the value of r.%3DQuestion Three(1) Find the particular solution for given cquations using separable variables method(i) y/ + y cos z = 4 cos z, y(0) = 0. (ii) (t² – 14t + 45)g/ = y, y(7) = 1.%3Ddy(iii)= sin z (cos 2y – cos² y), y(0) = 27. (iv) y':zy + 3z – y - 3%3Dy(0) = 1.%3Ddr%3Dry – 2x + 4y - 8'

Question

Please solve the attached questions (2) Use appropriate substitution to solve the following(i)dytan (x+y). (ii) y:1-1-y%3D%3Ddr(iii) y&#34; + (y/)² + 1= 0. (iv) xy&#34;+ (v/)² = 0.Question Four(1) In a third-order chemical reaction the number of grams X of compound obtained bycombining three chemicals is governed bydX= K(a - X)(8 – X)(-X).dtSolve the equation under the assumption a B #y.(2) The population P(t) at any time in Ras Al Khaimah is governed by the initial-valueproblemdP= 0.1P – 0.0004P?,dtP(0) = 200,where t measured in year. What is the limiting value of the population?. At what time willthe population be equal to one-half of this limiting valuc?.(3) In one model of the changing population P(t) of a community it is assumed thatdP dBdDdtdtdtwhere dB/dt and dD/dt are the birth and death rates; respectively.(i) Solve for P(t) ifdB= KP anddD= K2P.dt(ii) Analyse the cases K1 > K2, K1 = K2, K1 < K2.(4) When intcrest is compounded continuously the amount of moncy S incrcases at arate proportional to the amount present at any time: dS/dt =rS where r is the annual rateof interest.(a) Find the amount accrucd at the cnd of 5 ycars when AED 5000 is deposited in asavings account drawing 5% annual interest compounded continuously.(b) In how many years will the initial sum deposited be double? Question One(1) Determine the order of the given differential equations and state whether the equationis lincar or nonlincar (a >0 constant )dy(i)dr- sin*(z)y = cos (z)y + 1. (ii) 7ty – e'y' =t2 + 4adr(iii) zª + (a² – 2a)z-- tela-2)=. (iv) y&#34; = y² + cos(x).dt%3D(2) Find the values of r, if(i) / = 7x, y = rz². (i) y&#34; – 7y&#34; + 6y/ = 0, y = e&#34;.(iii) ty&#34; + 8ty/ + 6y = 0, y = t&#34;.Question Two(1) Show that z = A cosh(wx)+ B sinh(wr) satisfies the differential equation z&#34;-w²r = 0.(2) given that y(t) = ae' + bet with y(0) = 2 and y/(0) = 2, find the values of a and b.%3D(3) The general solution to second-order differential equation 5y&#34; = 2y is in the formy(x) = ce&#34; + C2, where c and cz are constants. Find the value of r.%3DQuestion Three(1) Find the particular solution for given cquations using separable variables method(i) y/ + y cos z = 4 cos z, y(0) = 0. (ii) (t² – 14t + 45)g/ = y, y(7) = 1.%3Ddy(iii)= sin z (cos 2y – cos² y), y(0) = 27. (iv) y':zy + 3z – y - 3%3Dy(0) = 1.%3Ddr%3Dry – 2x + 4y - 8'

Accepted Answer

As per our guideline we are supposed to answer only one asked question with first three subparts.…

(2) Use appropriate substitution to solve the following 1-1-y dy (i) dr tan (x+ v). (ii) y = %3D z+y (iii) y" + (y')² + 1 = 0. (iv) zy" + (y/)² = 0.