14.7.3 Approximate solution of differential equations

Problem: Find the solution to the boundary value problem using the finite difference method. \[ \begin{array}{l} y^{\prime \prime}+q(x) y^{\prime}=f(x), \quad x \in[a ; b] \\ y(a)=y_{a}, \quad y(b)=y_{b} \end{array} \] a) with the step \( h=\frac{b-a}{5} \); b) with the step \( h=\frac{b-a}{10} \); c) estimate the error by the Runge rule; d) plot the graphs of the approximate solutions. \[ a=0, b=1 \text {. } \]