Solve yn yn1 003yn1 007 y0 y0Solutiongiven yn yn1 003y

Solve y_n - y_n-1 = 0.03y_n-1 - 0.07, y(0) = y_0

given yn - y(n-1) = 0.03y(n-1) -0.07 ------>equation1

now plug n=n-1 in the above equation

we get y(n-1) -y(n-2) =0.03(n-2) -0.07 ----->equation2

now subtract equation2 from equation 1

y(n-1) -y(n-2) -0.03(n-2) +0.07 - -[ yn - y(n-1) - 0.03y(n-1) +0.07 ]=0

y(n-1) -1.03y(n-2) +0.07 -yn +1.03y(n-1) -0.07 =0

2.03y(n-1) -1.03y(n-2) -yn=0

now let us say yn = x^n

2.03 x^n-1 -1.03 x^n-2 -x^n =0

x^n [ 2.03/x -1.03/x^2 -1]=0

[2.03x -1.03 -x^2]=0

x^2 -2.03x +1.03=0

x^2 -1x -1.03x+1.03=0

x(x-1) -1.03(x-1)=0

(x-1)(x-1.03)=0

x= 1 or x=1.03

then yn = A(1)^n +B(1.03)^n

we have to find out A,B value from y(0) value

plug n=0

y0 = A+B

y1 = 1.03y0 -0.07

plug n=1

1.03y0 -0.07 = A+1.03B

solve and get A and B value in terms of y0