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1.3E: Direction Fields for First Order Equations (Exercises)

• • Contributed by William F. Trench
• Andrew G. Cowles Distinguished Professor Emeritus (Mathamatics) at Trinity University

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In Exercises 1–11 a direction field is drawn for the given equation. Sketch some integral curves. Figure $$\PageIndex{6}$$: A direction field for $$y'= {\frac{x}{y}}$$ Figure $$\PageIndex{7}$$: A direction field for $${y'= {2xy^2\over1+x^2}}$$ Figure $$\PageIndex{1}$$: A direction field for $$y'=x^2(1+y^2)$$ Figure $$\PageIndex{1}$$: A direction field for $$y'= {1\over1+x^2+y^2}$$ Figure $$\PageIndex{1}$$: A direction field for $$y'=-(2xy^2+y^3)$$ Figure $$\PageIndex{1}$$: A direction field for $$y'=(x^2+y^2)^{1/2}$$ Figure $$\PageIndex{1}$$: A direction field for $$y'=\sin xy$$ Figure $$\PageIndex{1}$$: A direction field for $$y'=e^{xy}$$ Figure $$\PageIndex{1}$$: A direction field for $$y'=(x-y^2)(x^2-y)$$ Figure $$\PageIndex{1}$$: A direction field for $$y'=x^3y^2+xy^3$$ Figure $$\PageIndex{1}$$: A direction field for $$y'=\sin(x-2y)$$

In Exercises [exer:1.3.12}-[exer:1.3.22} construct a direction field and plot some integral curves in the indicated rectangular region.

[exer:1.3.12] $$y'=y(y-1); \quad \{-1\le x\le 2,\ -2\le y\le2\}$$

[exer:1.3.13] $$y'=2-3xy; \quad \{-1\le x\le 4,\ -4\le y\le4\}$$

[exer:1.3.14] $$y'=xy(y-1); \quad \{-2\le x\le2,\ -4\le y\le 4\}$$

[exer:1.3.15] $$y'=3x+y; \quad \{-2\le x\le2,\ 0\le y\le 4\}$$

[exer:1.3.16] $$y'=y-x^3; \quad \{-2\le x\le2,\ -2\le y\le 2\}$$

[exer:1.3.17] $$y'=1-x^2-y^2; \quad \{-2\le x\le2,\ -2\le y\le 2\}$$

[exer:1.3.18] $$y'=x(y^2-1); \quad \{-3\le x\le3,\ -3\le y\le 2\}$$

[exer:1.3.19] $$y'= {x\over y(y^2-1)}; \quad \{-2\le x\le2,\ -2\le y\le 2\}$$

[exer:1.3.20] $$y'= {xy^2\over y-1}; \quad \{-2\le x\le2,\ -1\le y\le 4\}$$

[exer:1.3.21] $$y'= {x(y^2-1)\over y}; \quad \{-1\le x\le1,\ -2\le y\le 2\}$$

[exer:1.3.22] $$y'=- {x^2+y^2\over1-x^2-y^2}; \quad \{-2\le x\le2,\ -2\le y\le 2\}$$

[exer:1.3.23] By suitably renaming the constants and dependent variables in the equations $T' = -k(T-T_m) \eqno{\rm(A)}$

and $G'=-\lambda G+r \eqno{\rm(B)}$ discussed in Section 1.2 in connection with Newton’s law of cooling and absorption of glucose in the body, we can write both as $y'=- ay+b, \eqno{\rm(C)}$ where $$a$$ is a positive constant and $$b$$ is an arbitrary constant. Thus, (A) is of the form (C) with $$y=T$$, $$a=k$$, and $$b=kT_m$$, and (B) is of the form (C) with $$y=G$$, $$a=\lambda$$, and $$b=r$$. We’ll encounter equations of the form (C) in many other applications in Chapter 2.

Choose a positive $$a$$ and an arbitrary $$b$$. Construct a direction field and plot some integral curves for (C) in a rectangular region of the form $\{0\le t\le T,\ c\le y\le d\}$

of the $$ty$$-plane. Vary $$T$$, $$c$$, and $$d$$ until you discover a common property of all the solutions of (C). Repeat this experiment with various choices of $$a$$ and $$b$$ until you can state this property precisely in terms of $$a$$ and $$b$$.

[exer:1.3.24] By suitably renaming the constants and dependent variables in the equations $P'=aP(1-\alpha P) \eqno{\rm(A)}$

and $I'=rI(S-I) \eqno{\rm(B)}$ discussed in Section 1.1 in connection with Verhulst’s population model and the spread of an epidemic, we can write both in the form $y'=ay-by^2, \eqno{\rm(C)}$ where $$a$$ and $$b$$ are positive constants. Thus, (A) is of the form (C) with $$y=P$$, $$a=a$$, and $$b=a\alpha$$, and (B) is of the form (C) with $$y=I$$, $$a=rS$$, and $$b=r$$. In Chapter 2 we’ll encounter equations of the form (C) in other applications..

Choose positive numbers $$a$$ and $$b$$. Construct a direction field and plot some integral curves for (C) in a rectangular region of the form $\{0\le t\le T,\ 0\le y\le d\}$

of the $$ty$$-plane. Vary $$T$$ and $$d$$ until you discover a common property of all solutions of (C) with $$y(0)>0$$. Repeat this experiment with various choices of $$a$$ and $$b$$ until you can state this property precisely in terms of $$a$$ and $$b$$.

Choose positive numbers $$a$$ and $$b$$. Construct a direction field and plot some integral curves for (C) in a rectangular region of the form $\{0\le t\le T,\ c\le y\le 0\}$

of the $$ty$$-plane. Vary $$a$$, $$b$$, $$T$$ and $$c$$ until you discover a common property of all solutions of (C) with $$y(0)<0$$.

You can verify your results later by doing Exercise 2.2. [exer:2.2.27}.

IN THIS CHAPTER we study first order equations for which there are general methods of solution.

SECTION 2.1 deals with linear equations, the simplest kind of first order equations. In this section we introduce the method of variation of parameters. The idea underlying this method will be a unifying theme for our approach to solving many different kinds of differential equations throughout the book.

SECTION 2.2 deals with separable equations, the simplest nonlinear equations. In this section we introduce the idea of implicit and constant solutions of differential equations, and we point out some differences between the properties of linear and nonlinear equations.

SECTION 2.3 discusses existence and uniqueness of solutions of nonlinear equations. Although it may seem logical to place this section before Section 2.2, we presented Section 2.2 first so we could have illustrative examples in Section 2.3.

SECTION 2.4 deals with nonlinear equations that are not separable, but can be transformed into separable equations by a procedure similar to variation of parameters.

SECTION 2.5 covers exact differential equations, which are given this name because the method for solving them uses the idea of an exact differential from calculus.

SECTION 2.6 deals with equations that are not exact, but can made exact by multiplying them by a function known called integrating factor.

29

A first order differential equation is said to be linear if it can be written as

$\label{eq:2.1.1} y'+p(x)y=f(x).$

A first order differential equation that can’t be written like this is nonlinear. We say that Equation \ref{eq:2.1.1} is homogeneous if $$f\equiv0$$; otherwise it is nonhomogeneous. Since $$y\equiv0$$ is obviously a solution of the homgeneous equation $y'+p(x)y=0,$

we call it the trivial solution. Any other solution is nontrivial.

[example:2.1.1] The first order equations \begin{aligned} x^2y'+3y&=&x^2,\\[2\jot] xy'-8x^2y&=&\sin x,\\ xy'+(\ln x)y&=&0,\\ y'&=&x^2y - 2,\end{aligned}

are not in the form Equation \ref{eq:2.1.1}, but they are linear, since they can be rewritten as \begin{aligned} y'+{3\over x^2}y&=&1,\\ y'-8xy&=&{\sin x\over x},\\ y'+{\ln x\over x}y&=&0,\\ y'-x^2y&=&-2.\end{aligned}

[example:2.1.2] Here are some nonlinear first order equations: