Question

In problems 1-4 Use Euler’s method to approximate the solution to the given initial value problem at the points $\mathbf{x}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1}\mathbf{,}\mathbf{0}\mathbf{.}\mathbf{2}\mathbf{,}\mathbf{0}\mathbf{.}\mathbf{3}\mathbf{,}\mathbf{0}\mathbf{.}\mathbf{4}$, and $\mathbf{0}\mathbf{.}\mathbf{5}$, using steps of size $\mathbf{0}\mathbf{.}\mathbf{1}\left(\mathbf{h}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{1}\right)$.

$\frac{\mathbf{dy}}{\mathbf{dx}}\mathbf{=}\frac{\mathbf{-}\mathbf{x}}{\mathbf{y}}$,$\mathbf{y}\mathbf{\left(}\mathbf{0}\mathbf{\right)}\mathbf{=}\mathbf{4}$

Short answer

$x_n$	0.1	0.2	0.3	0.4	0.5
$y_n$	4	3.998	3.992	3.985	3.975

Step-by-step solution

Write the recursive formula

One has, $f\left(x,y\right)=\frac{-x}{y}, x_0=0, y_0=4, h=0.1$.

Then$$\begin{gathered} y_{n+1}=y_n+h.f\left(x_n,y_n\right) \\ =y_n+\left(0.1\right)\left(\frac{-x_n}{y_n}\right) \end{gathered}$$

Put  n=0 to find  role="math" localid="1663928029230" y1

Now, find the value of $y_1$

$$\begin{gathered} y_2=y_1+\left(0.1\right)\left(\frac{-x_1}{y_1}\right) \\ =4+\left(0.1\right)\left(\frac{-0.1}{4}\right) \\ =4-0.0025 \\ =3.998 \end{gathered}$$

Thus, the value is ${\mathbf{y}}_{\mathbf{2}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{998}$ when ${\mathbf{x}}_{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{2}$

Put  n=2 to find  y3

Now the value of $y_3$

$$\begin{gathered} y_3=y_2+\left(0.1\right)\left(\frac{-x_2}{y_2}\right) \\ =3.9975+\left(0.1\right)\left(\frac{-0.2}{3.9975}\right) \\ =3.9975-0.005 \\ =3.992 \end{gathered}$$

So, the values get ${\mathit{y}}_{\mathbf{3}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{992}$ when ${\mathit{x}}_{\mathbf{3}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{3}$

Put  n=3 to find  y4

The value of $y_4$is

$$\begin{gathered} y_4=y_3+(0.1)\left(\frac{-x_3}{y_3}\right) \\ =3.9925+(0.1)\left(\frac{-0.3}{3.9925}\right) \\ =3.9925-0.0075 \\ =3.985 \end{gathered}$$

Consequently, the value of ${\mathbf{y}}_{\mathbf{4}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{985}$ when ${\mathbf{x}}_{\mathbf{4}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{4}$

Put  n=4 to find  y5

The value of $y_5$is

$$\begin{gathered} y_5=y_4+\left(0.1\right)\left(\frac{-x_4}{y_4}\right) \\ =3.9849+\left(0.1\right)\left(\frac{-0.4}{3.9849}\right) \\ =3.9849-0.0100 \\ =3.975 \end{gathered}$$

Therefore, the value of ${\mathbf{y}}_{\mathbf{5}}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{975}$when ${\mathbf{x}}_{\mathbf{5}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{5}$.

Therefore, the solution is

localid="1663929244756" ${\mathit{x}}_{\mathbf{n}}$	0.1	0.3	0.4	0.5	0.6
localid="1663929266027" ${\mathit{y}}_{\mathbf{n}}$	4	3.998	3.992	3.985	3.975