Paper 2, Section I, C

Differential Equations | Part IA, 2017

(a) The numbers z1,z2,z_{1}, z_{2}, \ldots satisfy

zn+1=zn+cn(n1),z_{n+1}=z_{n}+c_{n} \quad(n \geqslant 1),

where c1,c2,c_{1}, c_{2}, \ldots are given constants. Find zn+1z_{n+1} in terms of c1,c2,,cnc_{1}, c_{2}, \ldots, c_{n} and z1z_{1}.

(b) The numbers x1,x2,x_{1}, x_{2}, \ldots satisfy

xn+1=anxn+bn(n1),x_{n+1}=a_{n} x_{n}+b_{n} \quad(n \geqslant 1),

where a1,a2,a_{1}, a_{2}, \ldots are given non-zero constants and b1,b2,b_{1}, b_{2}, \ldots are given constants. Let z1=x1z_{1}=x_{1} and zn+1=xn+1/Unz_{n+1}=x_{n+1} / U_{n}, where Un=a1a2anU_{n}=a_{1} a_{2} \cdots a_{n}. Calculate zn+1znz_{n+1}-z_{n}, and hence find xn+1x_{n+1} in terms of x1,b1,,bnx_{1}, b_{1}, \ldots, b_{n} and U1,,UnU_{1}, \ldots, U_{n}.

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