Chapter 6

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

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

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 \(t=0\). $$ \begin{aligned} &x^{\prime}=-y+t \\ &y^{\prime}=x-t \\ &x(0)=-3, y(0)=5 \end{aligned} $$

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

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

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 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 \(t=0\). $$ \begin{aligned} &x^{\prime}=6 x+y+6 t \\ &y^{\prime}=4 x+3 y-10 t+4 \\ &x(0)=0.5, y(0)=0.2 \end{aligned} $$

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

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

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

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 \(t=0\). $$ \begin{aligned} &x^{\prime}+4 x-y^{\prime}=7 t \\ &x^{\prime}+y^{\prime}-2 y=3 t \\ &x(0)=1, y(0)=-2 \end{aligned} $$

Consider the initial-value problem \(y^{\prime}=(x+y-1)^{2}, y(0)=2\). Use the improved Euler's method with \(h=0.1\) and \(h=0.05\) to obtain approximate values of the solution at \(x=0.5 .\) At each step compare the approximate value with the exact value of the analytic solution.

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

The electrostatic potential \(u\) between two concentric spheres of radius \(r=1\) and \(r=4\) is determined from $$ \frac{d^{2} u}{d r^{2}}+\frac{2}{r} \frac{d u}{d r}=0, \quad u(1)=50, u(4)=100 $$ Use the method of this section with \(n=6\) to approximate the solution of this boundary-value problem.

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

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 \(t=0\). $$ \begin{aligned} &x^{\prime}+y^{\prime}=4 t \\ &-x^{\prime}+y^{\prime}+y=6 t^{2}+10 \\ &x(0)=3, y(0)=-1 \end{aligned} $$

Although it may not be obvious from the differential equation, its solution could "behave badly" near a point \(x\) at which we wish to approximate \(y(x) .\) Numerical procedures may give widely differing results near this point. Let \(y(x)\) be the solution of the initial-value problem \(y^{\prime}=x^{2}+y^{3}\), \(y(1)=1\) (a) Use a numerical solver to obtain the graph of the solution on the interval \([1,1.4]\). (b) Using the step size \(h=0.1\), compare the results obtained from Euler's method with the results from the improved Euler's method in the approximation of \(y(1.4)\).

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

If air resistance is proportional to the square of the instantaneous velocity, then the velocity \(v\) of a mass \(m\) dropped from a given height \(h\) is determined from $$ m \frac{d v}{d t}=m g-k v^{2}, k>0 $$ Let \(v(0)=0, k=0.125, m=5\) slugs, and \(g=32 \mathrm{ft} / \mathrm{s}^{2}\). (a) Use the \(\mathrm{RK} 4\) method with \(h=1\) to approximate the velocity \(v(5)\). (b) Use a numerical solver to graph the solution of the IVP on the interval \([0,6]\). (c) Use separation of variables to solve the IVP and then find the actual value \(v(5)\)

Consider the initial-value problem \(y^{\prime}=2 y, y(0)=1\). The analytic solution is \(y=e^{2 x}\). (a) Approximate \(y(0.1)\) using one step and Euler's method. (b) Find a bound for the local truncation error in \(y_{1}\). (c) Compare the actual error in \(y_{1}\) with your error bound. (d) Approximate \(y(0.1)\) using two steps and Euler's method. (e) Verify that the global truncation error for Euler's method is \(O(h)\) by comparing the errors in parts (a) and (d).

Consider the boundary-value problem $$ y^{\prime \prime}+x y=0, \quad y^{\prime}(0)=1, \quad y(1)=-1 $$ (a) Find the difference equation corresponding to the differential equation. Show that for \(i=0,1,2, \ldots, n-1\), the difference equation yields \(n\) equations in \(n+1\) unknowns \(y_{-1}, y_{0}, y_{1}, y_{2}, \ldots, y_{n-1}\). Here \(y_{-1}\) and \(y_{0}\) are unknowns since \(y_{-1}\) represents an approximation to \(y\) at the exterior point \(x=-h\) and \(y_{0}\) is not specified at \(x=0\). (b) Use the central difference approximation (5) to show that \(y_{1}-y_{-1}=2 h\). Use this equation to eliminate \(y_{-1}\) from the system in part (a). (c) Use \(n=5\) and the system of equations found in parts (a) and (b) to approximate the solution of the original boundary-value problem.

A mathematical model for the area \(A\) (in \(\mathrm{cm}^{2}\) ) that a colony of bacteria \((B .\) dendroides \()\) occupies is given by $$ \frac{d A}{d t}=A(2.128-0.0432 A) $$ Suppose that the initial area is \(0.24 \mathrm{~cm}^{2}\). (a) Use the \(\mathrm{RK} 4\) method with \(h=0.5\) to complete the following table. $$ \begin{array}{|l|lllll|} \hline t \text { (days) } & 1 & 2 & 3 & 4 & 5 \\ \hline A \text { (observed) } & 2.78 & 13.53 & 36.30 & 47.50 & 49.40 \\ \hline A \text { (approximated) } & & & & & \\ \hline \end{array} $$ (b) Use a numerical solver to graph the solution of the initialvalue problem. Estimate the values \(A(1), A(2), A(3), A(4)\), and \(A(5)\) from the graph. (c) Use separation of variables to solve the initial-value problem and compute the values \(A(1), A(2), A(3), A(4)\), and \(A(5)\)

Consider the boundary-value problem \(y^{\prime \prime}=y^{\prime}-\sin (x y)\), \(y(0)=1, y(1)=1.5\). Use the shooting method to approximate the solution of this problem. (The actual approximation can be obtained using a numerical technique, say, the fourth-order Runge-Kutta method with \(h=0.1\); even better, if you have access to a CAS, such as Mathematica or Maple, the NDSolve function can be used.)

Consider the initial-value problem \(y^{\prime}=2 y, y(0)=1\). The analytic solution is \(y(x)=e^{2 x}\) (a) Approximate \(y(0.1)\) using one step and the fourth-order RK4 method. (b) Find a bound for the local truncation error in \(y_{1}\). (c) Compare the actual error in \(y_{1}\) with your error bound. (d) Approximate \(y(0.1)\) using two steps and the \(\mathrm{RK} 4\) method. (e) Verify that the global truncation error for the \(\mathrm{RK} 4\) method is \(O\left(h^{4}\right)\) by comparing the errors in parts (a) and (d).

Consider the initial-value problem \(y^{\prime}=2 x-3 y+1, y(1)=5\). The analytic solution is $$ y(x)=\frac{1}{9}+\frac{2}{3} x+\frac{38}{9} e^{-3(x-1)} $$ (a) Find a formula involving \(c\) and \(h\) for the local truncation error in the \(n\)th step if the RK4 method is used. (b) Find a bound for the local truncation error in each step if \(h=0.1\) is used to approximate \(y(1.5)\). (c) Approximate \(y(1.5)\) using the RK4 method with \(h=0.1\) and \(h=0.05\). See Problem 3. You will need to carry more than six decimal places to see the effect of reducing the step size.

The RK4 method for solving an initial-value problem over an interval \([a, b]\) results in a finite set of points that are supposed to approximate points on the graph of the exact solution. In order to expand this set of discrete points to an approximate solution defined at all points on the interval \([a, b]\), we can use an interpolating function. This is a function, supported by most computer algebra systems, that agrees with the given data exactly and assumes a smooth transition between data points. These interpolating functions may be polynomials or sets of polynomials joined together smoothly. In Mathematica the command \(\mathbf{y}=\) Interpolation[data] can be used to obtain an interpolating function through the points data \(=\left\\{\left\\{x_{0}, y_{0}\right\\},\left\\{x_{1}, y_{1}\right\\}, \ldots,\left\\{x_{n}, y_{n}\right\\}\right\\} .\) The interpolating function \(\mathbf{y}[\mathbf{x}]\) can now be treated like any other function built into the computer algebra system. (a) Find the analytic solution of the initial-value problem \(y^{\prime}=-y+10 \sin 3 x ; y(0)=0\) on the interval \([0,2] .\) Graph this solution and find its positive roots. (b) Use the RK4 method with \(h=0.1\) to approximate a solution of the initial- value problem in part (a). Obtain an interpolating function and graph it. Find the positive roots of the interpolating function on the interval \([0,2]\).