91Ó°ÊÓ

Given the initial-value problems in Problems use the Runge-Kutta method with \(h=0.1\) to obtain a four-decimal-place approximation to the indicated value.\(y^{\prime}=x^{2}+y^{2}, y(0)=1 ; \quad y(0.5)\)

In Problems 1-10 use the finite difference method and the indicated value of \(n\) to approximate the solution of the given boundary-value problem. \(y^{\prime \prime}+5 y^{\prime}=4 \sqrt{x}, \quad y(1)=1, y(2)=-1 ; \quad n=6\)

In Problems 5-12 sketch-or use a computer to obtain - the direction field for the given differential equation. Indicate several possible solution curves. $$ y^{\prime}=x+y $$

Use the Runge-Kutta method to approximate \(x(0.2)\) and \(y(0.2)\). Compare the results for \(h=0.2\) and \(h=0.1\). $$ \begin{aligned} &x^{\prime}=2 x-y \\ &y^{\prime}=x \\ &x(0)=6, y(0)=2 \end{aligned} $$

In Problems 5-12 sketch-or use a computer to obtain - the direction field for the given differential equation. Indicate several possible solution curves. $$ y \frac{d y}{d x}=-x $$

Given the initial-value, use Euler's formula to obtain a four-decimal approximation to the indicated value. First use \(h=0.1\) and then use \(h=0.05\). $$ y^{\prime}=e^{-7}, y(0)=0 ; \quad y(0.5) $$

In Problems 1-10 use the finite difference method and the indicated value of \(n\) to approximate the solution of the given boundary-value problem. \(x^{2} y^{\prime \prime}+3 x y^{\prime}+3 y=0, \quad y(1)=5, y(2)=0 ; \quad n=8\)

In Problems 5-8 use the Adams-Bashforth/Adams-Moulton method to approximate \(y(1.0)\), where \(y(x)\) is the solution of the given initial-value problem. Use \(h=0.2\) and \(h=0.1\) and the Runge-Kutta method to compute \(y_{1}, y_{2}\), and \(y_{3}\). \(y^{\prime}=(x-y)^{2}, y(0)=0\)

Given the initial-value, use Euler's formula to obtain a four-decimal approximation to the indicated value. First use \(h=0.1\) and then use \(h=0.05\). $$ y^{\prime}=x+y^{2}, y(0)=0 ; \quad y(0.5) $$

Given the initial-value problems in Problems use the Runge-Kutta method with \(h=0.1\) to obtain a four-decimal-place approximation to the indicated value.\(y^{\prime}=x+y^{2}, y(0)=0 ; \quad y(0.5)\)