Use Euler's method with step sizes h=0.1, h=0.05, and h=0.025 to find approximate values of the solution of the initial-value problem y'+3y=7e^(4x), y(0)=2 at x=0, 0.1, 0.2, 0.3, ..., 1.0.

Question

anonymous · Answer

$$y'=f(x,y)=7e^{4x}-3y$$
Fill in the table using $y_{n+1}=y_n+hf(x_n,y_n)$ with $x_{n+1}=x_n+nh$ and $h=0.1$ for $0\le n\le10$. 
$$\begin{array}{c|c|c|c}
n&x_n&y_n&f(x_n,y_n)\
\hline
0&0&2&7e^0-3(2)=1\
1&0+0.1=0.1&2+0.1(1)=2.1&7e^{0.4}-3(2.1)\approx 4.14277\
2&0+2(0.1)=0.2&2.1+0.1(4.14277)\approx 2.51428&\cdots\
\vdots&\vdots&\vdots&\vdots\
10&0+10(0.1)=1&\cdots&\cdots
\end{array}$$
If you follow this format for $h=0.05$ and $h=0.025$, you'll obviously have to take more steps. Tedious? Yes.

idealist10 · Answer

@SithsAndGiggles So what are the answers?

anonymous · Answer

At $x=0$, the approximate solution is the initial condition, $y=0$.

At $x=0.1$, i.e. the first step using $h=0.1$, gives an approximation of $y=2.1$.

For this particular step size, each successive $y_n$ will be the approximate solutions the question wants.

anonymous · Answer

You can try writing a script/program to do the recurrence for you, but that takes some technical coding knowledge. There are a few of these available freely on the web. This one seems to work pretty well, and agrees with (at the very least) all the points I worked out. http://www.mathscoop.com/calculus/differential-equations/euler-method-calculator.php