Lesson: Integral Curves of Vector Fields

Mathematics

In this lesson, we will learn how to find the integral curve of a vector field.

Worksheet: 6 Questions

Q1:

The figures show the vector field ο‡³βˆ’π‘¦,π‘₯+52𝑦, together with several of its flows.

Suppose we know that for some numbers π‘˜ the integral curves π‘₯=𝑓(𝑑),𝑦=𝑔(𝑑) are such that 𝑓 and 𝑔 are linear combinations of some 𝑒. What are the values of π‘˜?

What are the parametric equations of the integral curve that is at (βˆ’1,0) when 𝑑=0?

What are the parametric equations of the integral curve that is at (0,2) when 𝑑=0?

What are the parametric equations of the integral curve that is at (βˆ’1,1) when 𝑑=0?

As π‘‘β†’βˆž and as π‘‘β†’βˆ’βˆž along an integral curve, the secant between (0,0) and (𝑓(𝑑),𝑔(𝑑)) approaches one of the lines 𝐿 and 𝐿 shown. What are the slopes of these two lines?

Q2:

The figures show the vector field 𝑦,π‘₯βˆ’32𝑦, together with several of its flows.

Suppose we know that, for some numbers π‘˜, the integral curves π‘₯=𝑓(𝑑) and 𝑦=𝑔(𝑑) are such that 𝑓 and 𝑔 are linear combinations of some 𝑒. What are the values of π‘˜?

What are the parametric equations of the integral curve that is at (0,2) when 𝑑=0?

What are the parametric equations of the integral curve that is at (βˆ’1,1) when 𝑑=0?

What are the parametric equations of the integral curve that is at (2,βˆ’2) when 𝑑=0?

Using the fact that 𝑒⋅𝑒=1ο‘‰οŽ‘οŠͺ, find a Cartesian equation satisfied by the points of the integral curve that is at (0,2) when 𝑑=0. You need not simplify your expression.

As π‘‘β†’βˆž and π‘‘β†’βˆ’βˆž along an integral curve, the secant between (0,0) and (𝑓(𝑑),𝑔(𝑑)) approaches one of the lines 𝐿 and 𝐿 shown. What are the slopes of these two lines?

Q3:

Consider the parametric curve π‘₯=𝑒(𝑏𝑑)cos, 𝑦=𝑒(𝑏𝑑)sin with constants π‘Ž and 𝑏. The figure shows the case π‘Ž=15 and 𝑏=5 for βˆ’πœ‹β‰€π‘‘β‰€2πœ‹.

Find a vector field such that the curve π‘₯=𝑒(𝑏𝑑)cos and 𝑦=𝑒(𝑏𝑑)sin is its integral curve.

Find a linear second-order differential equation satisfied by π‘₯.

You can check that π‘₯=𝑒(𝑏𝑑)sin is also a solution to this differential equation and therefore any π‘₯=𝑓(𝑑)=𝑃𝑒(𝑏𝑑)+𝑄𝑒(𝑏𝑑)cossin for constants 𝑃 and 𝑄. Using the vector field, determine the corresponding function 𝑦=𝑔(𝑑) so that π‘₯=𝑓(𝑑) and 𝑦=𝑔(𝑑) is an integral curve.

For the case π‘Ž=15 and 𝑏=5, find parametric equations for the integral curve that starts at the point (3,2) when 𝑑=0.

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