Approximation of $\sqrt2$ using Euler's method

ordinary-differential-equations

We exploit the fact that $x(t)=\sqrt t$ satisfies the given differential equation with initial condition $x(1)=1$. Using Euler's Method, we then get:

$$x\left(\frac32\right)\approx x(1)+\frac12\cdot x'(1)=1+\frac12\cdot\frac{1}{2\cdot1}=\frac54$$ $$x(2)\approx x\left(\frac32\right)+\frac12\cdot x'\left(\frac32\right)\approx \frac54 + \frac12\cdot\frac{1}{2\cdot\frac54}=\frac{29}{20}=1.45$$

Hence $\sqrt2=x(2)\approx1.45$

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