Given the initial value problem (y^{prime}=y-y^{2}, y(0)=0.5) : (a) Solve the problem exactly using standard analytical methods

Question:

Given the initial value problem $y^{\prime}=y-y^{2}, y(0)=0.5$ :

(a) Solve the problem exactly using standard analytical methods for first-order ordinary differential equations.

(b) Apply the Runge-Kutta method (of fourth order) and solve the problem for $0 \leq x \leq 1$, with a step size of $h=0.2$. Compute the error, using the exact values in part (a).

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