I am working through this paper by John Cochrane on Asset Pricing. In the paper he solves for the present value budget constraint of the agent by applying inverse operator methods to the flow budget constraint (on page 3):
$$k_t = \frac{1}{D - r} (y_t - c_t) $$
Where $D$ is the difference operator, and $k_t, y_t$, and $c_t$ are variables. From which he arrives at:
$$k_t = \int_{0}^{\infty} e^{-r\tau} y_{t + \tau} d \tau - \int_{0}^{\infty} e^{-r\tau} c_{t + \tau} d \tau $$
Now, from my understanding of the inverse operator (for example, such as discussed in response to this post), I arrive at the following:
$$k_t = e^{rt}\left[\int_{0}^{\infty} e^{-r\tau} y_{t + \tau} d \tau - \int_{0}^{\infty} e^{-r\tau} c_{t + \tau} d \tau\right] $$
Am I doing something wrong here? How does Cochrane arrive at the expression above?