Apply Eulers Method to the differential equation dBtdt ARt

Apply Eulers Method to the differential equation dB(t)/dt = - AR(t)/R(t) + 1 R(0) = 1s and develop a difference equation approximation (ie replace dR(t)/dt by the forward difference approximation and re-arrange. Using a time step of delta t = 1 second, estimate the rate of production of time 1, 2 & 3 seconds.

Solution

Euler method is y_n+1 = y_n + h y\'_n

here y_n^\'=- Ay_n/(y_n+1)    y₀ = 1; and step size h=1;

therefore y_{t+1 =}y_t - 1.Ay_t/(y_t +1 )

t=0 s

y_{1 =} =y₀ - 1.Ay₀/(y₀ +1 ) = 1 - A/2 = 1 - 0.5 A

t=1s

y_{2 =} =y₁ - 1.Ay₁/(y₁ +1 ) = 1 - 0.5 A + A*(1-0.5A)/(2 -0.5A) = 1 - 0.5A +A(1-0.5A)/(2 - 0.5 A)

= ( 2 - 1.5 A + 0.25 A² + A - 0.5 A² ) / (2 - 0.5 A ) = ( -0.25 A² - 0.5A + 2 ) / (2 - 0.5A )

t=2

Y₃ = y_{2 -} Ay₂/(1 +y₂ ) = ( -0.25 A² - 0.5A + 2 ) / (2 - 0.5A ) - A . ( -0.25 A² - 0.5A + 2 ) / (2 - 0.5A ) /(1+ ( -0.25 A² - 0.5A + 2 ) / (2 - 0.5A ) ) = ( -0.25 A² - 0.5A + 2 ) / (2 - 0.5A ) - A( -0.25 A² - 0.5A + 2 )/(4 - A - 0.25 A² )