91Ó°ÊÓ

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. The general solution of a differential equation is \(y=-4.9 x^{2}+C_{1} x+C_{2} .\) To find a particular solution, you must be given two initial conditions.

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. Slope fields represent the general solutions of differential equations.

The exact solution of the differential equation \(\frac{d y}{d x}=-2 y\) where \(y(0)=4\), is \(y=4 e^{-2 x}\). (a) Use a graphing utility to complete the table, where \(y\) is the exact value of the solution, \(y_{1}\) is the approximate solution using Euler's Method with \(h=0.1, y_{2}\) is the approximate solution using Euler's Method with \(h=0.2, e_{1}\) is the absolute error \(\left|y-y_{1}\right|, e_{2}\) is the absolute error \(\left|y-y_{2}\right|\), and \(r\) is the ratio \(e_{1} / e_{2}\). $$ \begin{array}{|l|l|l|l|l|l|l|} \hline \boldsymbol{x} & 0 & 0.2 & 0.4 & 0.6 & 0.8 & 1 \\ \hline \boldsymbol{y} & & & & & & \\ \hline \boldsymbol{y}_{1} & & & & & & \\ \hline \boldsymbol{y}_{2} & & & & & & \\ \hline \boldsymbol{e}_{\mathbf{1}} & & & & & & \\ \hline \boldsymbol{e}_{2} & & & & & & \\ \hline \boldsymbol{r} & & & & & & \\ \hline \end{array} $$ (b) What can you conclude about the ratio \(r\) as \(h\) changes? (c) Predict the absolute error when \(h=0.05\).

The diagram shows a simple electric circuit consisting of a power source, a resistor, and an inductor. A model of the current \(I\), in amperes (A), at time \(t\) is given by the first- order differential equation \(L \frac{d I}{d t}+R I=E(t)\) where \(E(t)\) is the voltage \((\mathrm{V})\) produced by the power source, \(R\) is the resistance, in ohms \((\Omega)\), and \(L\) is the inductance, in henrys (H). Suppose the electric circuit consists of a \(24-V\) power source, a \(12-\Omega\) resistor, and a 4 - \(\mathrm{H}\) inductor. (a) Sketch a slope field for the differential equation. (b) What is the limiting value of the current? Explain.