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I need help to solve the following difference equation: $$y_{t+1}-y_{t}+ty_{t+1}y_{t}=0$$

I start by dividing with $y_{t+1}y_{t}$. Then I get: $$y_{t}^{-1}-y_{t+1}^{-1}=-t$$ Then I assume $y_{t}=\frac{1}{m^{t}}$ and find general solution to corresponding homogeneous equation: $$y_t=C_{1}\left ( 1 \right )^{t}+C_{2}t(1)^{t}\quad\Rightarrow\quad y_t=C_{1}+C_{2}t$$ Hoping that the solution is correct so far, I need help in finding a particular solution to nonhomogeneous equation.

Thanks!

A.Γ.
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Dragan
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1 Answers1

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With $z_t=y_t^{-1}$ we get $$ z_{t+1}-z_t=t. $$ Summing up for $t=0,\ldots,n-1$ you will get the telescopic sum in LHS and an easy to calculate in RHS. It gives you $z_n$ and, thus, $y_n$.

P.S. Your homogeneous equation would be $z_{t+1}=z_t$, it has only constant solutions, where does the term $...+C_2t$ for $y_t$ come from?

A.Γ.
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