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

Q1:

The figures show the vector field ο³βπ¦,π₯+52π¦ο·, together with several of its flows.

Suppose we know that for some numbers π the integral curves π₯=π(π‘),π¦=π(π‘) are such that π and π are linear combinations of some ποο. What are the values of π?

What are the parametric equations of the integral curve that is at (β1,0) when π‘=0?

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

What are the parametric equations of the integral curve that is at (β1,1) when π‘=0?

As π‘ββ and as π‘βββ along an integral curve, the secant between (0,0) and (π(π‘),π(π‘)) approaches one of the lines πΏο§ and πΏο¨ shown. What are the slopes of these two lines?

Q2:

The figures show the vector field ο³π¦,π₯β32π¦ο·, together with several of its flows.

Suppose we know that, for some numbers π, the integral curves π₯=π(π‘) and π¦=π(π‘) are such that π and π are linear combinations of some ποο. What are the values of π?

What are the parametric equations of the integral curve that is at (2,β2) when π‘=0?

Using the fact that ο½ποβ π=1οο‘οͺο±ο¨ο, find a Cartesian equation satisfied by the points of the integral curve that is at (0,2) when π‘=0. You need not simplify your expression.

As π‘ββ and π‘βββ along an integral curve, the secant between (0,0) and (π(π‘),π(π‘)) approaches one of the lines πΏο§ and πΏο¨ shown. What are the slopes of these two lines?

Q3:

Consider the parametric curve π₯=π(ππ‘)οΊοcos, π¦=π(ππ‘)οΊοsin with constants π and π. The figure shows the case π=15 and π=5 for βπβ€π‘β€2π.

Find a vector field such that the curve π₯=π(ππ‘)οΊοcos and π¦=π(ππ‘)οΊοsin is its integral curve.

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

You can check that π₯=π(ππ‘)οΊοsin is also a solution to this differential equation and therefore any π₯=π(π‘)=ππ(ππ‘)+ππ(ππ‘)οΊοοΊοcossin for constants π and π. Using the vector field, determine the corresponding function π¦=π(π‘) so that π₯=π(π‘) and π¦=π(π‘) is an integral curve.

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

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy.