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December 5th, 2013, 10:04 PM  #1 
Newbie Joined: Nov 2013 Posts: 4 Thanks: 0  Solving Diff Eqns using Euler's and Midpoint formulas
the function y(x) satisfies the differential equation dy/dx=cos(xy) and the condition y(1)=2. a) Verify that the Euler Formula with step length 0.25 gives y(1.25)=1.89596 to 5 d.p. b) Use the midpoint formula with h=0.25 to obtain an estimate of the value of y(2). Giving answer to 3 d.p. OK I thought I had this sorted out because with previous questions I had no problem but my answer here is different than the answer my sheet tells me I should be getting, I'll show my working. a) Euler's Formula> yr+1=yr+hf(xr,yr) I know that h=0.25 and that x0=1 and y0=2 I also know that f(xr,yr)={cos(1)(2)} So for (x1,y1) I've done y1=2+0.25{cos(1)(2)} y1=1.895963291 or 1.89596 to 5 d.p. so part a) is done no problem And I now have a second set of coordinates that are (x1,y1) they are (1.25 , 1.895963291) so I can now use the Midpoint formula to find (x2,y2) b) Midpoint formula> yr+1=yr1+2hf(xr,yr) y2=y0+2hf(x1,y1) I know that y0=2 and that x1=1.25 and y1=1.895963291, also that h=0.25 y2=2+2(0.25){cos(1.25)(1.895963291)} y2=2.298919812 And I now I have a third set of coordinates that are (x2,y2) they are (1.5 , 2.298919812) y3=y1+2hf(x2,y2) I know that y1=1.895963291 and that x2=1.5 and y2=2.298919812, also that h=0.25 y3=1.895963291+2(0.25){cos(1.5)(2.298919812)} y3=1.977272868 And I now have a fourth set of coordinates that are (x3,y3) they are (1.75 , 1.97727286 y4=y2+2hf(x3,y3) I know that y2=2.298919812 and that x3=1.75 and y3=1.977272868, also that h=0.25 y4=2.298919812+2(o.25){cos(1.75)(1.97727286} y4=2.122699267 And I now have a fifth set of coordinates that are (x4,y4) they are (2 , 2.122699267) Now the question asks me to find the value of y(2) and I thought that is what I had done here with y(2)=2.122699267, but my answer sheet tells me that my answer should be 1.204 to 3d.p. Any help would be much appreciated as the previous questions that I had done my answer was correct, thanks in advance 

Tags 
diff, eqns, euler, formulas, midpoint, solving 
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