Question

In Exercises \(41-44,\) use Euler's Method with increments of \(\Delta x=0.1\) to approximate the value of \(y\) when \(x=1.3 .\) \(\frac{d y}{d x}=y-1\) and \(y=3\) when \(x=1\)

Short answer

The approximate value of \(y\) when \(x=1.3\) is \(y=3.662\)

Step-by-step solution

Identify the given values

The differential equation is \(\frac{d y}{d x}=y-1\) and the initial condition is \(x=1, y=3\). The increment \(\Delta x=0.1\). The value of \(y\) is desired when \(x=1.3\). That means 3 steps are needed as the increment is 0.1.

Apply Euler's Method for first iteration

Use the formula \(y_{new} = y_{old} + \Delta x \cdot f(x_{old}, y_{old})\), with \(f(x,y) = y - 1\). For the first iteration \(x=1, y=3\), the estimate of \(y\) at \(x=1.1\) is \(y_{new} = 3 + 0.1 \cdot (3 - 1) =3.2\).

Second and third iterations

Continue this process two more times. At \(x=1.1, y=3.2\), the estimate of \(y\) at \(x=1.2\) is \(y_{new} = 3.2 + 0.1 \cdot (3.2 - 1) = 3.42\). Then, at \(x=1.2, y=3.42\), the approximate value of \(y\) at \(x=1.3\) is \(y_{new} = 3.42 + 0.1 \cdot (3.42 - 1) =3.662\).

Result

After the third iteration, approximate the value of \(y\) when \(x=1.3\) to be \(y=3.662\)