Use Euler\'s method to calculate approximately y(3) if the function y(t) satisfi

Question

Use Euler's method to calculate approximately y(3) if the function y(t) satisfies the differential equation y' = y + y-1 and the initial condition y( 1) = 1. Use the step 0.5 and calculate with 4 decimal places.

Explanation / Answer

y'= y + 1/y, y(1) =1 AND h= 5 find y(3) =? 4 d.p. = four decimal points y(1) =1==> y(xo) = yo ==> xo= 1 and yo=1 y'= y + 1/y ==> y'=f(x,y) ==> f(x,y) = y + 1/y xo=1 x1=xo+h=1+0.5=1.5 x2=x1+h=x0+2h = 1.5+0.5=2 x3= x2+h= xo+3h= 2+0.5=2.5 x4= x3+h= xo+4h= 2.5+0.5=3 y(x1)= y(xo+h ) = y(xo) + h f(xo,yo) y(1.5)=y(1+0.5) = y(1)+ 0.5 ( y(1) + 1/y(1) ) = 1+ (1/2)( 1+1) = 1+1 =2.0000 y(1.5)= 2.0000 y(x2)= y(x1+h ) = y(x1) + h f(x1,y1) y(2)=y(1.5+0.5) = y(1.5)+ 0.5 ( y(1.5) + 1/y(1.5) ) =3.2500 y(2.5)= y(2)+ 0.5 ( y(2) + 1/y(2) ) =5.0289 y(3.5)= y(2.5)+ 0.5 ( y(2.5) + 1/y(2.5) ) = 7.6428 y(3)= y(3)+ 0.5 ( y(3) + 1/y(3) ) = 11.5296 y(4)= y(3.5)+ 0.5 ( y(3.5) + 1/y(3.5) ) = 17.3378

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