Chapter 6

Use the finite difference method and the indicated value of \(n\) to approximate the solution of the given boundary-value problem. $$ y^{\prime \prime}+9 y=0, \quad y(0)=4, y(2)=1 ; n=4 $$

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 \(h=0.05\) . \(y^{\prime}=2 x-3 y+1, \quad y(1)=5 ; y(1.5)\)

Use Euler's method to approximate \(y(0.2)\), where \(y(x)\) is the solution of the initial-value problem $$ y^{\prime \prime}-4 y^{\prime}+4 y=0, \quad y(0)=-2, y^{\prime}(0)=1 . $$ Use \(h=0.1\). Find the exact solution of the problem, and compare the actual value of \(y(0.2)\) with \(y_{2}\).

Use the RK4 method with \(h=0.1\) to approximate \(y(0.5)\), where \(y(x)\) is the solution of the initial-value problem \(y^{\prime}=(x+y-1)^{2}, y(0)=2\). Compare this approximate value with the actual value obtained in Problem 11 in Exercises 6.1.

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 \(h=0.05\). $$ y^{\prime}=2 x-3 y+1, \quad y(1)=5 ; y(1.5) $$

Use the finite difference method and the indicated value of \(n\) to approximate the solution of the given boundary-value problem. $$ y^{\prime \prime}-y=x^{2}, \quad y(0)=0, y(1)=0 ; n=4 $$

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

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

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

In Problems \(3-12\), use the \(\mathrm{RK} 4\) method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=2 x-3 y+1, \quad y(1)=5 ; y(1.5) $$

Use the Adams-Bashforth-Moulton method to approximate \(y(0.8)\), where \(y(x)\) is the solution of the given initial-value problem. Use \(h=0.2\) and the \(\mathrm{RK} 4\) method to compute \(y_{1}, y_{2}\), and \(y_{3}\). $$ y^{\prime}=2 x-3 y+1, \quad y(0)=1 $$

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

Construct a table comparing the indicated values of \(y(x)\) using Euler's method, the improved Euler's method, and the RK4 method. Compute to four rounded decimal places. Use \(h=0.1\) and then use \(h=0.05\). $$ y^{\prime}=\sqrt{x+y}, \quad y(0.5)=0.5 $$ \(y(0.6), y(0.7), y(0.8), y(0.9), y(1.0)\)

Use the RK4 method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=2 x-3 y+1, \quad y(1)=5 ; y(1.5) $$

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

Use the finite difference method and the indicated value of \(n\) to approximate the solution of the given boundary-value problem. $$ y^{\prime \prime}-10 y^{\prime}+25 y=1, \quad y(0)=1, y(1)=0 ; n=5 $$

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

Use the Adams-Bashforth-Moulton method to approximate \(y(0.8)\), where \(y(x)\) is the solution of the given initial-value problem. Use \(h=0.2\) and the \(\mathrm{RK} 4\) method to compute \(y_{1}, y_{2}\), and \(y_{3}\). $$ y^{\prime}=4 x-2 y, \quad y(0)=2 $$

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

Construct a table comparing the indicated values of \(y(x)\) using Euler's method, the improved Euler's method, and the RK4 method. Compute to four rounded decimal places. Use \(h=0.1\) and then use \(h=0.05\). $$ y^{\prime}=x y+y^{2}, \quad y(1)=1 $$ \(y(1.1), y(1.2), y(1.3), y(1.4), y(1.5)\)

Use the RK4 method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=4 x-2 y, \quad y(0)=2 ; y(0.5) $$

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

Use the finite difference method and the indicated value of \(n\) to approximate the solution of the given boundary-value problem. $$ y^{\prime \prime}-4 y^{\prime}+4 y=(x+1) e^{2 x}, \quad y(0)=3, y(1)=0 ; n=6 $$

Use the \(\mathrm{RK} 4\) method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=1+y^{2}, \quad y(0)=0 ; y(0.5) $$

Use the Adams-Bashforth-Moulton method to approximate \(y(1.0)\), where \(y(x)\) is the solution of the given initial-value problem. First use \(h=0.2\) and then use \(h=0.1\). Use the RK4 method to compute \(y_{1}, y_{2}\), and \(y_{3}\). $$ y^{\prime}=1+y^{2}, \quad y(0)=0 $$

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 \(h=0.05\) . \(y^{\prime}=e^{-y}, \quad y(0)=0 ; y(0.5)\)

Use the RK4 method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=1+y^{2}, \quad y(0)=0 ; y(0.5) $$

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

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

Use the \(\mathrm{RK} 4\) method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=x^{2}+y^{2}, \quad y(0)=1 ; y(0.5) $$

Use the Adams-Bashforth-Moulton method to approximate \(y(1.0)\), where \(y(x)\) is the solution of the given initial-value problem. First use \(h=0.2\) and then use \(h=0.1\). Use the RK4 method to compute \(y_{1}, y_{2}\), and \(y_{3}\). $$ y^{\prime}=y+\cos x, \quad y(0)=1 $$

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

Use the RK4 method with \(h=0.1\) to obtain a four-decimal approximation to the indicated value. $$ y^{\prime}=x^{2}+y^{2}, \quad y(0)=1 ; y(0.5) $$

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

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

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

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 \(t=0\). $$ \begin{aligned} &x^{\prime}=2 x-y \\ &y^{\prime}=x \\ &x(0)=6, y(0)=2 \end{aligned} $$

Use the Adams-Bashforth-Moulton method to approximate \(y(1.0)\), where \(y(x)\) is the solution of the given initial-value problem. First use \(h=0.2\) and then use \(h=0.1\). Use the RK4 method to compute \(y_{1}, y_{2}\), and \(y_{3}\). $$ y^{\prime}=(x-y)^{2}, \quad y(0)=0 $$

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

Use Euler's method with \(h=0.1\) to approximate \(x(0.2)\) and \(y(0.2)\), where \(x(t), y(t)\) is the solution of the initial-value problem $$ \begin{aligned} x^{\prime} &=x+y \\ y^{\prime} &=x-y \\ x(0) &=1, y(0)=2 \end{aligned} $$

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

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

Use the finite difference method and the indicated value of \(n\) to approximate the solution of the given boundary-value problem. $$ x^{2} y^{\prime \prime}-x y^{\prime}+y=\ln x, \quad y(1)=0, y(2)=-2 ; n=8 $$

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

Use the Adams-Bashforth-Moulton method to approximate \(y(1.0)\), where \(y(x)\) is the solution of the given initial-value problem. First use \(h=0.2\) and then use \(h=0.1\). Use the RK4 method to compute \(y_{1}, y_{2}\), and \(y_{3}\). $$ y^{\prime}=x y+\sqrt{y}, \quad y(0)=1 $$

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 \(t=0\). $$ \begin{aligned} &x^{\prime}=x+2 y \\ &y^{\prime}=4 x+3 y \\ &x(0)=1, y(0)=1 \end{aligned} $$

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

Use the finite difference method with \(n=10\) to approximate the solution of the boundary-value problem \(y^{\prime \prime}+6.55(1+x) y=1, y(0)=0, y(1)=0\).

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

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