Solve the differential equation $(\frac{dy}{dx}{)}^2-(e^x+e^{-x})\frac{dy}{dx}+1=0$.

Solution

1. The given differential equation is $(\frac{dy}{dx}{)}^2-(e^x+e^{-x})\frac{dy}{dx}+1=0$.
2. Factoring the quadratic expression in terms of $\frac{dy}{dx}$ gives $(\frac{dy}{dx}-e^x)(\frac{dy}{dx}-e^{-x})=0$.
3. This implies two separate first-order equations: $\frac{dy}{dx}=e^x$ and $\frac{dy}{dx}=e^{-x}$.
4. Integrating $\frac{dy}{dx}=e^x$ yields $y=e^x+C$, and integrating $\frac{dy}{dx}=e^{-x}$ yields $y=-e^{-x}+C$.
5. Hence the answer is (A).