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An ordinary differential equation $\text{(ODE)}, \: \dfrac{dy}{dx}=2y$, with an initial condition $y(0)=1$, has the analytical solution $y=e^{2x}$.

Using Runge-Kutta second order method, numerically integrate the $\text{ODE}$ to calculate $y$ at $x=0.5$ using a step size of $h=0.5$.

If the relative percentage error is defined as, $$\varepsilon = \begin{vmatrix}  \dfrac{y_{\text{analytical}} – y_{\text{numerical}}}{y_\text{analytical}} \end{vmatrix} \times 100$$ Then the value of $\varepsilon$ at $x=0.5$ is _________

  1. $0.06$
  2. $0.8$
  3. $4.0$
  4. $8.0$
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