91Ó°ÊÓ

Given the initial-value problems, use the improved Euler's method to obtain a four-decimal approximation to the indicated value. First use \(h=0.1\) and then use \(h=0.05\) . \(y^{\prime}=x y^{2}-\frac{y}{x}, \quad y(1)=1 ; y(1.5)\)

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

Use the \(\mathrm{RK} 4\) method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=(x-y)^{2}, \quad y(0)=0.5 ; y(0.5) $$

Given the initial-value, use the improved Euler's method to obtain a four- decimal approximation to the indicated value. First use \(h=0.1\) and then use \(h=0.05\). $$ y^{\prime}=x y^{2}-\frac{y}{x}, \quad y(1)=1 ; y(1.5) $$

Use the RK4 method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=(x-y)^{2}, \quad y(0)=0.5 ; y(0.5) $$

Use the Runge-Kutta method to approximate \(x(0.2)\) and \(y(0.2) .\) First use \(h=0.2\) and then use \(h=0.1\). Use a numerical solver and \(h=0.1\) to graph the solution in a neighborhood of \(t=0\). $$ \begin{aligned} &x^{\prime}=-y+t \\ &y^{\prime}=x-t \\ &x(0)=-3, y(0)=5 \end{aligned} $$

Use the \(\mathrm{RK} 4\) method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=x y+\sqrt{y}, \quad y(0)=1 ; y(0.5) $$

Use the finite difference method and the indicated value of \(n\) to approximate the solution of the given boundary-value problem. $$ y^{\prime \prime}+x y^{\prime}+y=x, \quad y(0)=1, y(1)=0 ; n=10 $$

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

Use the RK4 method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=x y+\sqrt{y}, \quad y(0)=1 ; y(0.5) $$