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The solution of the differential equation $k^{2} \frac{d^{2} y}{d x^{2}}=y-y_{2}$ under the boundary conditions **(i)** $y=y_{1}$ at $x=0$ and **(ii)** $y=y_{2}$ at $x=\infty$, where $k, y_{1}$ and $y_{2}$ are constants, is

- $y=\left(y_{1}-y_{2}\right) \exp \left(-x / k^{2}\right)+y_{2}$
- $y=\left(y_{2}-y_{1}\right) \exp (-x / k)+y_{1}$
- $y=\left(y_{1}-y_{2}\right) \sinh (x / k)+y_{1}$
- $y=\left(y_{1}-y_{2}\right) \exp (-x / k)+y_{2}$