Final answer:
By integrating s''(x) = e^x twice and using the initial conditions s(0)=1.2 and s(1)=5, we find the expression for s(x) as s(x) = e^x + (3.8 - e)x + 1.2 with no unknown constants.
Step-by-step explanation:
To find an expression for s(x) given that s''(x) = e^x, we need to integrate twice to find s'(x) and s(x). The first integration gives s'(x), and the second integration yields s(x) with some unknown constants that can be determined using the initial conditions provided (s(0)=1.2 and s(1)=5).
First integration of s''(x) results in s'(x) = e^x + C1, and second integration gives s(x) = e^x + C1x + C2. Using the initial conditions, we solve for C1 and C2. For s(0)=1.2, we get C2 = 1.2. Now applying s(1)=5, we get 5 = e^1 + C1 × 1 + 1.2, which simplifies to C1 = 5 - e - 1.2. Therefore, s(x) is s(x) = e^x + (5 - e - 1.2)x + 1.2, which simplifies to s(x) = e^x + (3.8 - e)x + 1.2, with no unknown constants.