Consider the following initial value problem  y 0 (t) = 3y − 2 cos(t) − 7 y(0.50) = 2 . An estimate of y(0.70), obtained by applying once the Taylor method of order 2 is: a. 1.508966975 b. 1.302834090 c. 0.1270954503 d. 1.468143997

To estimate y(0.70) using the Taylor method of order 2, we will first find y(0.70) using the initial condition and the given differential equation.

Given differential equation: y'(t) = 3y - 2\cos(t) - 7

Initial condition: y(0.50) = 2

To use the Taylor method of order 2, we need to find y(0.70) using the following steps:

Step 1: Find y'(t) at t = 0.50
y'(0.50) = 3(2) - 2\cos(0.50) - 7

Step 2: Find y''(t) at t = 0.50
y''(t) = \frac{dy'}{dt}
y''(0.50) = 3y'(0.50) + 2\sin(0.50)

Step 3: Use Taylor method of order 2 to find y(0.70)
y(0.70) \approx y(0.50) + h y'(0.50) + \frac{h^2}{2} y''(0.50)
y(0.70) \approx 2 + 0.20 \times y'(0.50) + \frac{0.20^2}{2} \times y''(0.50)

Now, substitute the values of y'(0.50) and y''(0.50) into the equation to find the estimate of y(0.70.

\text{Answer: } y(0.70) \approx 1.468143997 \, \text{(option d)}