1 point suppose that we use eulers method to approximate the...

Suppose that we use Euler's method to approximate the solution to the differential equation dy/dx = x^5y; y(0.1) = 7. Let f(x,y) = x^5/y. We let x0 = 0.1 and y0 = 7 and pick a step size h = 0.2. Euler's method is the following algorithm. From xn and yn, our approximations to the solution of the differential equation at the nth stage, we find the next stage by computing xn+1 = xn + h, yn+1 = yn + h*f(xn,yn). Complete the following table. Your answers should be accurate to at least seven decimal places. n xn yn 0 0.1 7 1 0.3 7.014 2 0.5 7.042 3 0.7 7.086 4 0.9 7.147 5 1.1 7.226 The exact solution can also be found using separation of variables. It is y(x) = (7*e^(x^6/6))/e^(1/6). Thus the actual value of the function at the point x = 1.1 is y(1.1) = (7*e^(1.1^6/6))/e^(1/6).

Transcript

00:01 Let's have a look at the question.

00:02 So here first we have been given that dy by dx is equals to x to the power 5 divided by y and it is given that y 0 .1 is equals to 7.

00:17 So now let us find the exact value for this given expression.

00:22 So here we have that x 0 is equals to 0 .1, y 0 is equals to 7 and h is equals to 0 .2.

00:32 So now using the formula that x n plus 1 is equals to x n plus h and y n plus 1 is equals to y n plus h multiplied with x n comma y n.

00:48 So now here we can write that y 1 is equals to y 0 plus h multiplied with f x 0 y 0.

00:59 So putting the values here we get 7 plus 0 .2 multiplied with 0 .1 to the whole power 5 divided by 7.

01:09 So this is equals to 7 .000000286.

01:16 Similarly the value for y 2 will be equals to 7 .000000286 plus 0 .2 multiplied with 0 .3 to the whole power 5 and divided by 7 .000000286.

01:38 So value for y 2 will be equals to 7 .000069715.

01:47 Now value for y 3 will be equals to 7 .000069715 and plus multiplied with 0 .2 and 0 .5 to the whole power 5 divided by 7 .000069715.

02:11 Now here the value for y 3 will be equals to 7 .0000 then we have 962563.

02:21 Now let us calculate the value for y 4 that is equals to 7 .0000962563 and plus 0 .2 multiplied with 0 .7 to the whole power 5 and divided by 7 .0000 and we have 962563.

02:49 So y 4 will be equals to 7 .005763903.

02:57 Now the value for y 4 is equals to 7 .005763903 plus 0 .2 multiplied with 0 .9 to the whole power 5 and divided by 7 .005763903.

03:17 So here we will get that the value for y 5 is equals to 7 .022621165.

03:26 We will see that the value for x n that is x 0 was 0 .1 then the value for x 1 was 0 .3 x 2 was 0 .5 then x 3 was 0 .7 x 4 was 0 .9 and x 5 was 1 .1...

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