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Mathematics LibreTexts

1.2: Slope fields

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    The general first order equation we are studying looks like

    \[ y' = f(x,y)\]

    In general, we cannot simply solve these kinds of equations explicitly. It would be nice if we could at least figure out the shape and behavior of the solutions, or if we could find approximate solutions. At this point it may be good to first try the Lab I and/or Project I from the IODE website:

    1.2.1 Slope fields

    As you have seen in IODE Lab I (if you did it), the equation \( y' = f(x,y) \) gives you a slope at each point in the \((x,y)\)- plane. We can plot the slope at lots of points as a short line through the point \((x,y)\) with the slope \(f(x,y).\) See Figure 1.1.


    Figure 1.1: Slope field of \( y' = xy \).

    We call this picture the slope field of the equation. If we are given a specific initial condition \( y(x_0)=y_0\), we can look at the location \( (x_0, y_0) \) and follow the slopes (see Figure 1.2).


    Figure 1.2: Slope field of \( y' = xy \) with a graph of solutions satisfying \( y(0) = 0.2\), \( y(0) = 0\), and \(y(0) = -0.2\).

    By looking at the slope field we can get a lot of information about the behavior of solutions. For example, in Figure 1.2 we can see what the solutions do when the initial conditions are \(y(0) > 0\), \( y(0) = 0\) and \( y(0) < 0 \). Note that a small change in the initial condition causes quite different behavior. On the other hand, plotting a few solutions of the equation \(y' = -y\), we see that no matter what \(y(0)\) is, all solutions tend to zero as \(x\) tends to infinity (see Figure 1.3).


    Figure 1.3: Slope field of \( y' = -y\) with a graph of a few solutions.

    1.2.2 Existence and uniqueness

    We wish to ask two fundamental questions about the problem

    \[ y' = f(x,y) \]


    \[ y(x_0) = y_0 \]

    (i) Does a solution exist?
    (ii) Is the solution unique (if it exists)?

    What do you think is the answer? The answer seems to be yes to both does it not? Well, pretty much. But there are cases when the answer to either question can be no.

    Since generally the equations we encounter in applications come from real life situations, it seems logical that a solution always exists. It also has to be unique if we believe our universe is deterministic. If the solution does not exist, or if it is not unique, we have probably not devised the correct model. Hence, it is good to know when things go wrong and why.

    Example \(\PageIndex{1}\):

    Attempt to solve

    \[ y' = \dfrac {1}{x}, ~ y(0) = 0.\]

    Integrate to find the general solution \( y = \ln \left \vert x \right \vert + C\). Note that the solution does not exist at \( x = 0 \). See Figure 1.4.


    Figure 1.4: Slope field of \(y' = \dfrac{1}{x}.\)


    Figure 1.5: Slope field of \( y' = 2 \sqrt{|y|}\) with two solutions satisfying \( y(0)=0.\)

    Example \(\PageIndex{2}\):


    \[ y' = 2 \sqrt {\left \vert y \right \vert}, ~~ y(0) = 0 .\]


    Figure 1.5: Slope field of \( y' = 2 \sqrt {\left \vert y \right \vert}\) with two solutions satisfying \( y(0) = 0 \).

    Note that \( y = 0 \) is a solution. But another solution is the function

    \[ y(x) =\begin {cases} x^2 & {\rm{if~}} x \ge 0 \\ -x^2 & {\rm{if~}} x < 0 \end {cases} \]

    It is hard to tell by staring at the slope field that the solution is not unique. Is there any hope? Of course there is. We have the following theorem, known as Picard’s theorem1.

    Theorem 1.2.1. (Picard’s theorem on existence and uniqueness). If \( f(x,y) \) is continuous (as afunction of two variables) and \(\dfrac{\partial f}{\partial y}\) exists and is continuous near some \( (x_0,y_0)\), then a solution to

    \[ y' = f(x,y)\]



    exists (at least for some small interval of \(x\)’s) and is unique.

    Note that the problems \(y' = \dfrac{1}{x}\), \( y(0) = 0 \) and \( y' = 2 \sqrt {\left \vert y \right \vert}\), \( y(0) = 0 \) do not satisfy the hypothesis of the theorem. Even if we can use the theorem, we ought to be careful about this existence business. It is quite possible that the solution only exists for a short while.

    Example \(\PageIndex{3}\):

    For some constant \(A\), solve:

    \( y' = y^2\), \(y(0) = A\).

    We know how to solve this equation. First assume that \( A \ne 0\), so \(y\) is not equal to zero at least for some \(x\) near 0. So \( x' = \dfrac{1}{y^2}\), so \(x = -\dfrac{1}{y} + C\), so \(y = \dfrac {1}{C - x}\). If \( y(0) = A \), then \( C = \dfrac{1}{A}\) so

    \[ y = \dfrac {1}{\dfrac{1}{A} - x}\]

    If \( A = 0\), then \( y = 0 \) is a solution.

    For example, when \(A = 1\) the solution “blows up” at \(x = 1\). Hence, the solution does not exist for all \(x\) even if the equation is nice everywhere. The equation \(y' = y^2\) certainly looks nice.

    For most of this course we will be interested in equations where existence and uniqueness holds, and in fact holds “globally” unlike for the equation \(y' = y^2\).

    1Named after the French mathematician Charles Émile Picard (1856 – 1941)