Euler's Method

Calculus Level 5

Given the differential equation $\frac{dy}{dx} = ye^{\pi x}$ determine an estimate for the value of $y(1)$ given that $y(0) = 3$ . Use Euler's method to determine your estimate, with a stepsize of $0.2$ . If your estimate is equal to $E$ , the solution to this question will be $\lfloor E \rfloor$ .

The answer is 67.

2 solutions

Agnishom Chattopadhyay
Jun 5, 2014

Here is some Mathematica Code:

euler[f_, {x_, x0_, xn_}, {y_, y0_}, steps_] := 
 Block[{xold = x0, yold = y0, sollist = {{x0, y0}}, x, y, h}, 
  h = N[(xn - x0)/steps];
  Do[xnew = xold + h;
   ynew = yold + h*(f /. {x -> xold, y -> yold});
   sollist = Append[sollist, {xnew, ynew}];
   xold = xnew;
   yold = ynew, {steps}];
  Return[sollist]]

Because stepsize is 0.2. The number of steps is $\frac{(1-0)}{0.2}=\boxed{5}$

Key in this:

euler[y E^(\[Pi] x), {x, 0, 1}, {y, 3}, 5]

You will get:

{{0, 3}, {0.2, 3.6}, {0.4, 4.94961}, {0.6, 8.42778}, {0.8, 
  19.529}, {1., 67.7471}}

So, the value of y at 1 is 67.7471

Nguyen Thanh Long
May 25, 2014

Apply Euler method: $y(t+h)=y(t)+h \times y(t) \times f'(t) \Rightarrow y(0.2 \times k)=y(0.2 \times (k-1)) \times [1+0.2 \times e^{0.2 \times \pi}]$ $\Rightarrow y(0.2)=3.6$ $y(0.4)=3.6 \times [1+0.2 * e^{0.2 \times \pi}]=4.9496$ $E=y(1)=67.747, \lfloor E \rfloor = \boxed{67}$

Sir, I haven't worked on the Euler's method ..But won't the usual method to this Problem give the solution ? y(1) = e^[(e^pi)/pi] ??

Suchit Kulkarni - 7 years ago

The question requires a numeric approximation of the ordinary differential equation. Euler's method is a simple explicit method for the given numerical analysis. Your 'usual' method doesn't readily provide a numerical approximation of the differential equation. Hence you need Euler's method to obtain an answer manually.

hamlet sentit - 6 years, 1 month ago

Euler's Method

The answer is 67.

2 solutions

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