Question

Please help :(

Answer 1

dy/dx = 5x
Use Euler's method to approximate y when x = 1 with 4 steps

Answer 2

Euler's method states that \[
y_{k+1}=y_k+h\times f(x_k,y_k)
\]

Answer 3

information missing: \(y(0)=?\)

Answer 4

step size: \[h = \frac{b-a}{n}={1\over4}=0.25\]

Answer 5

y(0) = 5!!!

Answer 6

and \[f(x,y)=5x\]

Answer 7

ok.. we start from k=0
0)
\[k=0\\
x_0=0\qquad y_0=5\\
y_1=y_0+h\times f(x_0,y_0)=5+0.25\times(5\times0)\\
x_1=x_0+h=0+0.25=0.25
\]

Answer 8

1)\[
k=1\\
y_2=y_1+h\times f(x_1,y_1)=5+0.25\times(5\times0.25)=5.3125\\
x_2=x_1+h=0.25+0.25=0.5
\]

Answer 9

3)\[
k=2\\
y_3=y_2+h\times f(x_2,y_2)=5.3125+0.25\times(5\times0.5)=5.9375\\
x_3=x_2+h=0.5+0.25=0.75
\]

Answer 10

and you continue two more times

Answer 11

how did you find the step size? if it says like 2 steps then would the h value be 1/2 ?

Answer 12

no.
step size = (final x - initial x) / number of steps

Answer 13

oh..didn't you get 1/4 by doing 1-0 / 4?

Answer 14

yep

Answer 15

oh ok. but do you know what it means if a problem says to solve a DE with an initial condition for [0,1] using 10 subintervals? how would you find the h value then?

Answer 16

what it means is
1) (0,1) -> \(y(x=0)=1\)
2) n = 10 -> number of steps

Answer 17

but it already says the initial condition is y(0) = 5?? ?? sorry
the whole problem basically says use euler's method to solve this DE with y(0) = 5, for [0,1] using 10 subintervals so what does that meannn

Answer 18

then the ew h becomes\[h=(1-0)/10=0.1\]
and find \[y_1,y_2,\ldots,y_{10}\]

Answer 19

Note that the exact solution for the given DE is: \(\Large y(x)={5\over2}(x^2+2)\)
the correct \(y(1)=7.5\)
the smaller the step size, the accurate your answer would be.