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.\(y^{\prime}=4 x-2 y, y(0)=2 ; \quad y(0.5)\)

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 \(h=0.05\). $$ y^{\prime}=4 x-2 y, y(0)=2 ; \quad y(0.5) $$

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.\(y^{t}=1+y^{2}, y(0)=0 ; \quad y(0.5)\)

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 \(y_{3}\). \(y^{\prime}=1+y^{2}, \quad y(0)=0\)

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. \(y^{\prime \prime}-4 y^{\prime}+4 y=(x+1) e^{2 x}, \quad y(0)=3, y(1)=0 ; \quad n=6\)

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

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 \(h=0.05\). $$ y^{\prime}=1+y^{2}, y(0)=0, \quad y(0.5) $$

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 \(h=0.05\). $$ y^{\prime}=x^{2}+y^{2}, y(0)=1 ; \quad y(0.5) $$

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.\(y^{\prime}=x^{2}+y^{2}, y(0)=1 ; \quad y(0.5)\)

Construct a table comparing the indicated values of \(y(x)\) using the Euler, improved Euler, and Runge-Kutta methods. Compute to four rounded decimal places. Use \(h=0.1\) and \(h=0.05\). $$ \begin{aligned} &y^{\prime}=x y+y^{2}, y(1)=1 ; \\ &y(1.1), y(1.2), y(1.3), y(1.4), y(1.5) \end{aligned} $$