(1 point) Suppose that we use Euler's method to approximate the solution to the differential equation

dy/dx=x^5/y; y(0.4)=7

Let . f(x,y)=x5/y.

We let x0=0.4x0=0.4 and y0=7y0=7 and pick a
step size h=0.2.h=0.2. Euler's method is the the
following algorithm. From xnxn and ynyn, 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).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.

nn | xnxn | ynyn |

00 | 0.40.4 | 77 |

11 | ||

22 | ||

33 | ||

44 | ||

55 |

The exact solution can also be found using separation of
variables. It is

y(x)=y(x)=

Thus the actual value of the function at the
point x=1.4x=1.4

y(1.4)=y(1.4)= .

Community Answer

{:[(dy)/(dx)=(x^(5))/(y)","y(0:4)=7","h=0.2],[f(x","y)=(x^(5))/(y)","x_(0)=0.4","y_(0)=7","h=0.2],[x_(1)=x_(0)+h=0.6","x_(2)=x_(1)+h=0.8","x_(3)=x_(2)+h=1","x_(4)=x_(3)+h=1-2_(3)],[x_(5)=x_(4)+h=1.4]:}Euber's formula is given by y_(n+1)=y_(n)+hf(x_(n),y_(n)){:[n=0],[y_(1)=y_(0)+hf(x_(0),y_(0))=y_(0)+h[(x_(0)^(5))/(y_(0))]=7+(0.2)[((0.4)^(5))/(7)]]:}{:[=7+(0.2)[(0.01024)/(7)]=7+(0.2)(0.001462857143)],[y_(1)=7.000292571;*;],[y(0.6)=7.000292571],[y_(2)=y_(1)+hf(x_(1),y_(1))=y_(1)+h[(x_(1)^(5))/(y_(1))]=7.000292571+(0.2)[((0.6)^(5))/(7.00292571)]],[n=1],[=7.000292571+(0.2)[(0.07776)/(7.000292571)]],[=7.000292571+(0.2)[0.011108107]=7.000292571+0.002221621131],[y_(2)=7.002514192" i.e; "],[y(0.8)=7.002514192],[n=2],[y_(3)=y_(2)+hf(x_(2),y_(2))=y_(2)+h[(x_(2)5)/(y_(2))]=2.002514192+(0.2)[((0.8)^(5))/(7.002514192)]],[=7.002514192+(0.2)[(0.32768)/(7.002514192)]],[=7.002514192+(0.2)(0.0467946211=7.002514192+0.009358924267],[y_(3)=7.011873116" i.e; "],[y(1)=2.011873116]:}{:[n=3],[y_(4)=y_(3)+hf(x_(3),y_(3))=y_(3)+h[(x_(3)^(5))/(y_(3))]=7.011873116+(0.2)[((1)^(5))/(2.011873116 ... See the full answer