Question

The given family of functions is the general solution of the differential equation on the indicated interval. Find a member of the family that is a solution of the initial-value problem.
y = c1ex + c2e−x, (−∞, ∞); y'' − y = 0, y(0) = 0, y'(0) = 7

Answer 1

Answer on question #85724 – Math – Differential Equations

Let $y(x) = C_1 e^x + C_2 e^{-x}$. The condition of $y^{\wedge} - y = 0$ is satisfied for all $x$: $y^{\wedge} = C_1 e^x - C_2 e^{-x}$, $y^{\wedge \wedge} = C_1 e^x + C_2 e^{-x} = y$.

From a condition of $y(0) = 0$ we receive $C_2 = -C_1$ and from $y^{\wedge}(0) = 7$ we receive $2C_1 = 7$. So, $C_1 = 3.5$, $C_2 = -3.5$, hence $y(x) = 3.5 e^x - 3.5 e^{-x}$.

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