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I am asked to solve the following problem. Use Euler's modified method to solve the initial value problem $\frac{dx}{dt} =\frac{1+x^2}{t} ,1\le t \le4, x(1)=0$

The step size is not given here. Now, I can solve this using Euler's method but I am not able to find the formula for modified one in the mentioned textbook. So, I need the method here. I need to use the following as textbook: Brain Bradie, A friendly Introduction to Numerical Analysis (Pearson)

Kavita
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    If we have an ODE $\frac{dx}{dt} = f(t,x)$ then IIRC the modified Euler's method says that $x_{n+1} = x_n + \frac{h}{2}\left[ f(t_n, x_n) + f(t_{n+1}, x_{n} + h f(t_n, x_n))\right]$ where $h = \Delta t$ is the step-size. – Winther Nov 15 '16 at 16:01
  • Can you post picture or source from where you got this method please. – Kavita Nov 15 '16 at 16:33
  • It's from memory so I can't guarantee that this is the same method your book calls Euler's modified method. However if we do a simple Google Search then the three top hits all agree with this defintion. – Winther Nov 15 '16 at 16:35
  • Yes, now I got the method in textbook under the heading "The most common second-order Runge-Kutta methods" – Kavita Nov 15 '16 at 16:37

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You can find the method under the name Heuns's method, also as (explicit) trapezoidal method.

k1 = h*f( t  , x    )
k2 = h*f( t+h, x+k1 )

y += 0.5 * ( k1 + k2 )
Lutz Lehmann
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