Let's refine the solution using the precise values you provided for each step. We'll use Euler's method to estimate f(2) and round the final result to three significant figures.

The differential equation is:

\frac{\differentialD y}{\differentialD x} = y e^x

The initial condition is f(0) = 1 .

Set up Euler's method:

Euler's method is given by:

y_{n+1} = y_n + h \cdot f(x_n, y_n)

where h = 0.5 .

Calculate the values step-by-step:

Step 1: From x = 0 to x = 0.5

Initial values: x_0 = 0 , y_0 = 1 .
Calculate f(x_0, y_0) = y_0 e^{x_0} = 1 \cdot e^0 = 1 .
Update using Euler's method:

y_1=y_0+h\cdot f(x_0,y_0)=1+0.5\cdot1=1.5

x_1 = x_0 + h = 0 + 0.5 = 0.5

Step 2: From x = 0.5 to x = 1.0

Initial values: x_1 = 0.5 , y_1 = 1.5 .
Calculate f(x_1, y_1) = y_1 e^{x_1} = 1.5 \cdot e^{0.5} .
Using a calculator, e^{0.5} \approx 1.6487 .
Thus, f(x_1, y_1) = 1.5 \cdot 1.6487 = 2.47305 .
Update using Euler's method:

y_2 = y_1 + h \cdot f(x_1, y_1) = 1.5 + 0.5 \cdot 2.47305 = 2.736525

x_2 = x_1 + h = 0.5 + 0.5 = 1.0

Step 3: From x = 1.0 to x = 1.5

Initial values: x_2 = 1.0 , y_2 = 2.736525 .
Calculate f(x_2, y_2) = y_2 e^{x_2} = 2.736525 \cdot e^{1.0} .
Using a calculator, e^{1.0} \approx 2.7183 .
Thus, f(x_2, y_2) = 2.736525 \cdot 2.7183 = 7.4343 .
Update using Euler's method:

y_3 = y_2 + h \cdot f(x_2, y_2) = 2.736525 + 0.5 \cdot 7.4343 = 6.453675

x_3 = x_2 + h = 1.0 + 0.5 = 1.5

Step 4: From x = 1.5 to x = 2.0

Initial values: x_3 = 1.5 , y_3 = 6.453675 .
Calculate f(x_3, y_3) = y_3 e^{x_3} = 6.453675 \cdot e^{1.5} .
Using a calculator, e^{1.5} \approx 4.4817 .
Thus, f(x_3, y_3) = 6.453675 \cdot 4.4817 = 28.917 .
Update using Euler's method:

y_4 = y_3 + h \cdot f(x_3, y_3) = 6.453675 + 0.5 \cdot 28.917 = 20.912175

x_4 = x_3 + h = 1.5 + 0.5 = 2.0

Rounding to three significant figures, the final result is f\left(2\right)\approx\boxed{20.9}.

Problem Bank - Differential Equations

Problem Bank - Differential Equations