Question

Using Euler’s Method with a step size of h=0.5, estimate the value for y(3) for the differential equation xy dy/dx= 1+ln(x^2), with y(2)=5. a. 5.352 b. 5.230 c. 5.238 d. 5.233

Answer 1

Euler's method uses the recurrence relation

$y_{n+1}=y_n+hf(x_n,y_n)$

to approximate the value of the solution $y(x)$ to the ODE $y'=f(x,y)$.

$xy\dfrac{\mathrm dy}{\mathrm dx}=1+\ln(x^2)\implies y'=\dfrac{1+\ln(x^2)}{xy}=f(x,y)$

With a step size of $h=0.5$, there will only be two steps necessary to find the approximate value of $y(3)$ based on the initial point $y(2)=5$. See the attached table below for the computation results.

To demonstrate how the table is generated: Since $y(2)=5$, you are using $(x_0,y_0)=(2,5)$.

$y_1=y_0+hf(x_0,y_0)$
$y_1=5+0.5\times\dfrac{1+\ln(2^2)}{2\times5}$
$y_1\approx5.1193$

The next point then uses $x_1=x_0+0.5=2.5$

$y_2=y_1+hf(x_1,y_1)$
$y_1=y_1+0.5\times\dfrac{1+\ln(2.5^2)}{2.5y_1}$
$y_2\approx5.230$