14.7.2 Approximate solution of differential equations

Problem: Numerically solve the Cauchy problem on the segment of length 0,8 for a first-order ordinary differential equation \( \left\{\begin{array}{l}y^{\prime}=f(x, y) \\ y(a)=y_{a}\end{array}\right. \), with the step \( h=0,2 \) : a) by the explicit Euler method with an error estimate by the Runge rule; b) by one of Runge-Kutta methods of the \( 2^{\text {nd }} \) order with an error estimate by the Runge rule; c) find the exact solution to the problem.