A calibration curve y(x) of a sensor satisfies y'' + y' - 6y = 0…

Step 1. The conditions. $y' = 2Ae^{2x} - 3Be^{-3x}$. So $A + B = 7$ and $2A - 3B = -1$.

Step 2. Solve. Multiply the first equation by $3$ and add: $3A + 3B + 2A - 3B = 21 - 1$, so $5A = 20$ and $A = 4$. Then $B = 7 - 4 = 3$.

Step 3. Evaluate. $y(1) = 4e^{2} + 3e^{-3} = 4 \times 7.389056 + 3 \times 0.049787 = 29.7056$.