Question

2. (10 points) Consider the initial value problem y = y-2. and y(1) = 0. (a) (4 points) Use Euler's method with step size 0.5 to approximate y(2) (i.e., the value of y when x = 2)? (b) (6 points) Solve the differential equation (with the specified initial value) to get y as a function of r.

Answer 1

I TOXI md given dy = y-2x & yeis=0 Aim o use Euleo 18 method with step size ha to approximate 712) solution - Now comparing with dy = f(x,y) . 4 (26)=Y = f(x,y)=4-9x and Xo=1, 40=0. h=0.5 . By Euler 8 method Ynti = ynth fixen, 4). - © y = yothf (80%) from @ y = 0 +0.5 ( 90-2x0) y = 0 +0.5 10-2x1) = 15*(-2) y = -1 1 In=o. ... My = Hoth = 17 0.5 = 1.5 . 12,= 1.51 | 4y=-11 Then second approximation n=d. from @ Hg = yeth f 12,14). 42 = 4th ( 49-981) 42 = -17 0.5 (-1-2x1.5). y = -1 +0.5 (-1-3) = -1-2 Ty2=-3

Az = x,+2h 212 = 1+2x(0.5). = 1+1. x2 = 2; y = -31 : put n=2, lucynth f 112,42). Fromegn % = 4 th (92-242) 49,= -3+(0.5) ( - 3.- 2x2) 43 = -3 + (0.51-7) yg = -3 -3.5 B0 22=2 4:.-6.5. // Ang

=y-au with mitial condition yui) = 0 solution - y = -2x. the This is linear first order of Now composing with dy + P(x) y = Q(x) P (x) = -1 , Q(x)= -9 = e. Then . If = espxldX S-ldx ex . | I-F = ēx] Then required solution l.fx yen) = / P-F *Q(x) dx +c + X 4x) = [ * (2x) x + c ē X Y ») = -9 [ Sxe *dx :] +c. ē yen) =-[-x ē* + Sēxdx] + c ẽ*4 (x) = -9 [-x + ^– 8*] + exyex) = 2 [xe* + ex] +c Ty(x) = 2 (4+1) + Cex Now lesing you=0, we have y (1) = 9 (1+1) +ce 0 = 4 + Cle) ! c=-46 I yex) = 2(x+1)4 eXt Then Ace TUCCE