Which of the following gives the solution of the difference equation
Yt = bYt - 1 + c, With initial condition, Y0 = a?
$Y _ { t } = b Y _ { t } - 1 + c$, with initial condition, $Y _ { 0 } = a$ ?
A) $\frac { c ( 1 - a + c ) b ^ { t } + c } { 1 - b }$

B) $\frac { c \left( 1 - b ^ { t } \right) } { 1 - b } + a b ^ { t }$

C) $\frac { a ( 1 - c ) b ^ { t } + a b } { 1 - b }$

D) $\frac { c b ^ { t } } { 1 - b } + a ( 1 - b )$

E) $\frac { a ( 1 - b + c ) b ^ { t } + c } { 1 - b }$

Question

Which of the following gives the solution of the difference equation 
Yt = bYt - 1 + c, With initial condition, Y0 = a?
 $Y _ { t } = b Y _ { t } - 1 + c$, with initial condition, $Y _ { 0 } = a$ ?
A) $\frac { c ( 1 - a + c ) b ^ { t } + c } { 1 - b }$

B) $\frac { c \left( 1 - b ^ { t } \right) } { 1 - b } + a b ^ { t }$

C) $\frac { a ( 1 - c ) b ^ { t } + a b } { 1 - b }$

D) $\frac { c b ^ { t } } { 1 - b } + a ( 1 - b )$

E) $\frac { a ( 1 - b + c ) b ^ { t } + c } { 1 - b }$

Quizplus · Accepted Answer

The correct solution to the given difference equation $Y_t = bY_{t-1} + c$ with initial condition $Y_0 = a$ is found by solving the difference equation iteratively or using the method of characteristic equations. The solution involves both the homogeneous part ($a b^t$, which accounts for the initial condition and the behavior of the system without the constant term c) and the particular solution ($\frac{c(1 - b^t)}{1 - b}$, which accounts for the constant term c). The combination of these two parts gives the complete solution, which correctly matches option B.

Which of the Following Gives the Solution of the Difference Yt=bYt−1+cY _ { t } = b Y _ { t } - 1 + cYt​=bYt​−1+c

Which of the Following Gives the Solution of the Difference $Y _ { t } = b Y _ { t } - 1 + c$