Use Euler's method to calculate the first three approximations to the given initial value problem for the specified increment size. Round your results to four decimal places.
-$$\mathrm { y } ^ { \prime } = \mathrm { y } - \mathrm { e } ^ { \mathrm { x } } - 3 , \mathrm { y } ( 1 ) = 1 , \mathrm { dx } = 0.5$$
A) $\mathrm { y } _ { 1 } = - 1.3591 , \mathrm { y } _ { 2 } = - 6.9796 , \mathrm { y } _ { 3 } = - 14.4639$
B) $\mathrm { y } _ { 1 } = - 1.3591 , \mathrm { y } _ { 2 } = - 5.1796 , \mathrm { y } _ { 3 } = - 13.5639$
C) $y _ { 1 } = - 1.3591 , y _ { 2 } = - 5.7796 , y _ { 3 } = - 13.8639$
D) $\mathrm { y } _ { 1 } = - 1.3591 , \mathrm { y } _ { 2 } = - 6.3796 , \mathrm { y } _ { 3 } = - 14.1639$

Question

Quizplus · Accepted Answer

Using Euler's method with the given differential equation and initial condition, the first three approximations are calculated as follows:  1. $ y_1 = y_0 + dx \cdot (y_0 - e^x - 3) = 1 + 0.5 \cdot (1 - e^1 - 3) = -1.3591 $  2. $ y_2 = y_1 + dx \cdot (y_1 - e^{x+dx} - 3) = -1.3591 + 0.5 \cdot (-1.3591 - e^{1.5} - 3) = -5.7796 $  3. $ y_3 = y_2 + dx \cdot (y_2 - e^{x+2dx} - 3) = -5.7796 + 0.5 \cdot (-5.7796 - e^2 - 3) = -13.8639 $

Use Euler's Method to Calculate the First Three Approximations to the Given