# In this problem, $$\displaystyle{y}={c}_{{1}}{e}^{{x}}+{c}_{{2}}{e}^{{-{x}}}$$ is a two-parameter

In this problem, $y={c}_{1}{e}^{x}+{c}_{2}{e}^{-x}$ is a two-parameter family of solutions of the second-order DE $y{}^{″}-y=0$. Find a solution of the second-order IVP consisting of this differential equation and the given initial conditions.
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Kingston Lowery

Solution:
has solution $y={c}_{1}\cdot {e}^{x}+{c}_{2}\cdot {e}^{-x}$
where ${c}_{1}$ and ${c}_{2}$ are constants
so, $y\left(0\right)={c}_{1}+{c}_{2}=1$ (i)
and; ${y}^{\prime }\left(x\right)={c}_{1}{e}^{x}-{c}_{2}{e}^{-x}$
${y}^{\prime }\left(0\right)={c}_{1}-{c}_{2}=8$ (ii)
so, (i)+(ii)
(i)-(ii)
so the solution is:
$y=\frac{9}{2}\cdot {e}^{x}-\frac{7}{2}\cdot {e}^{-x}$

Jeffrey Jordon