Approximate solutions of diff. equations/Approximate…

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Problem: Find a solution to the boundary value problem, using the finite difference method. \( \left\{\begin{array}{c}-y^{\prime \prime}+q(x) y=f(x) \\ y(0)=y_{0}, \quad y(1)=y_{1}\end{array} \quad\right. \) with steps \( h_{1}=\frac{1}{3}, \quad h_{2}=\frac{1}{6} \) and estimate the error by the Runge rule. Plot the graphs of the obtained approximate solutions. \begin{tabular}{|c|c|c|c|c|} \hline \( \mathrm{N} \) & \( q(x) \) & \( f(x) \) & \( y_{0} \) & \( y_{1} \) \\ \hline 11 & 1 & \( 1+6 x-x^{3} \) & 1 & 0 \\ \hline \end{tabular}

14.7.1 Approximate solution of differential equations

6.43 $

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.

14.7.2 Approximate solution of differential equations

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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. \[ a=0, b=1 \text {. } \]

14.7.3 Approximate solution of differential equations

9 $