WebWork 17.4 #2

Problem: Consider the vector field $\vec{v}=-4x\vec{i}+4y\vec{j}$. (a) Find the system of differential equations associated with this vector field. (b) Solve the system you found above to find the flow.

Ideas:¬†For part (a), you’re just going to take the derivatives as we did in class. E.g., for my numbers I would have $x’=-4x$ and $y’=4y$. This makes part (b) a bit problematic, and you’re going to need to remember some of your differential equations tricks. The idea is that I want a function $x(t)$ such that $x’=-4x$. Notice that the rate of change of $x$ is a function containing $x$, and so you should automatically be thinking that $x$ is an exponential function (and the same is true for $y$). In fact, if we set $x(t)=ae^{ct}$ (for constants $a$ and $c$) then we have

\[x'=-4x=-4ae^{ct}\]

Of course, if $x=ae^{ct}$ then $x’=cae^{ct}$, and so $ca=-4a$ and $c=-4$. Thus, we have $x(t)=ae^{-4t}$, where $a$ simply provides an initial condition (a specific curve in the slope field). Similarly, we can find $y(t)$ given that $y’=4y$ and $y(t)=be^{dt}$.

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