1. The direction field for dy/dx = 4x/y is shown in Figure 1.12. (a) Verify that the straight lines y = ±2x are solution curves, provided x ? 0. (b) Sketch the solution curve with initial condition y(0) = 2. (c) Sketch the solution curve with initial condition y(2) = 1. (d) What can you say about the behavior of the above solutions as x ? +?? How about x ? -??
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The direction field for $d y / d x=4 x / y$ is shown in Figure $1.12 .$ $$\begin{array}{l}{\text { (a) Verify that the straight lines } y=\pm 2 x \text { are solution }} \\ {\text { curves, provided } x \neq 0 \text { . }} \\ {\text { (b) Sketch the solution curve with initial condition }} \\ {y(0)=2 .} \\ {\text { (c) Sketch the solution curve with initial condition }} \\ {y(2)=1 .} \\ {\text { (d) What can you say about the behavior of the above }} \\ {\text { solutions as } x \rightarrow+\infty ? \text { How about } x \rightarrow-\infty ?}\end{array}$$
Introduction
Direction Fields
Consider the differential equation dy/dt = t^2 - y whose direction field is shown at right. (a) Find the family of solutions to this differential equation. (b) Find the solution to the initial value problem with y(-2) = 2, and add this solution curve to the direction field. (c) Give an equation for the curve that all solutions y(t) approach as t approaches infinity, and add this solution to the direction field.
Adi S.
Consider the differential equation dy/dx = (y - y^2)/x for all x ≠ 0. (a) Verify that y = x/(x + C), x ≠ -C and C ≠ 0 is a general solution for the given differential equation and show that all solutions contain (0, 0). (b) Write an equation of the particular solution that contains the point (1, 2), and find the value of dy/dx at (0, 0) for this solution. (c) Write an equation of the vertical and horizontal asymptotes of the particular solution found in (b). (d) The slope field for the given differential equation is provided. Sketch both branches of the particular solution curve that passes through the point (1, 2).
Sri K.
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