Lesson: Integral Curves of Vector Fields Mathematics

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

Lesson Worksheet

Q1:

The figures show the vector field 𝑦,𝑥+52𝑦, together with several of its flows.

Suppose we know that for some numbers 𝑘 the integral curves 𝑥=𝑓(𝑡),𝑦=𝑔(𝑡) are such that 𝑓 and 𝑔 are linear combinations of some 𝑒. What are the values of 𝑘?

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

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

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

As 𝑡 and as 𝑡 along an integral curve, the secant between (0,0) and (𝑓(𝑡),𝑔(𝑡)) 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,𝑥32𝑦, together with several of its flows.

Suppose we know that, for some numbers 𝑘, the integral curves 𝑥=𝑓(𝑡) and 𝑦=𝑔(𝑡) are such that 𝑓 and 𝑔 are linear combinations of some 𝑒. What are the values of 𝑘?

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

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

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

Using the fact that 𝑒𝑒=1, find a Cartesian equation satisfied by the points of the integral curve that is at (0,2) when 𝑡=0. You need not simplify your expression.

As 𝑡 and 𝑡 along an integral curve, the secant between (0,0) and (𝑓(𝑡),𝑔(𝑡)) approaches one of the lines 𝐿 and 𝐿 shown. What are the slopes of these two lines?

Q3:

Consider the parametric curve 𝑥=𝑒(𝑏𝑡)cos, 𝑦=𝑒(𝑏𝑡)sin with constants 𝑎 and 𝑏. The figure shows the case 𝑎=15 and 𝑏=5 for 𝜋𝑡2𝜋.

Find a vector field such that the curve 𝑥=𝑒(𝑏𝑡)cos and 𝑦=𝑒(𝑏𝑡)sin is its integral curve.

Find a linear second-order differential equation satisfied by 𝑥.

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

For the case 𝑎=15 and 𝑏=5, find parametric equations for the integral curve that starts at the point (3,2) when 𝑡=0.

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