Worksheet: Integral Curves of Vector Fields

In this worksheet, we will practice finding the integral curve of a vector field.

Q1:

The figures show the vector field 𝑦 , 𝑥 + 5 2 𝑦 , 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 ?

  • A 1 2 and 3
  • B 1 3 and 2
  • C 1 3 and 3
  • D 1 2 and 2
  • E 1 4 and 2

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

  • A 𝑥 = 4 3 𝑒 + 1 3 𝑒 , 𝑦 = 2 3 𝑒 + 2 3 𝑒
  • B 𝑥 = 4 3 𝑒 1 3 𝑒 , 𝑦 = 2 3 𝑒 + 2 3 𝑒
  • C 𝑥 = 4 3 𝑒 + 1 3 𝑒 , 𝑦 = 2 3 𝑒 2 3 𝑒
  • D 𝑥 = 4 3 𝑒 + 1 3 𝑒 , 𝑦 = 2 3 𝑒 2 3 𝑒
  • E 𝑥 = 4 3 𝑒 + 1 3 𝑒 , 𝑦 = 2 3 𝑒 + 2 3 𝑒

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

  • A 𝑥 = 4 3 𝑒 + 4 3 𝑒 , 𝑦 = 2 3 𝑒 + 8 3 𝑒
  • B 𝑥 = 4 3 𝑒 + 4 3 𝑒 , 𝑦 = 2 3 𝑒 + 8 3 𝑒
  • C 𝑥 = 4 3 𝑒 4 3 𝑒 , 𝑦 = 2 3 𝑒 8 3 𝑒
  • D 𝑥 = 4 3 𝑒 4 3 𝑒 , 𝑦 = 2 3 𝑒 + 8 3 𝑒
  • E 𝑥 = 4 3 𝑒 4 3 𝑒 , 𝑦 = 2 3 𝑒 8 3 𝑒

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

  • A 𝑥 = 2 3 𝑒 + 𝑒 3 , 𝑦 = 𝑒 3 + 2 3 𝑒
  • B 𝑥 = 2 3 𝑒 𝑒 3 , 𝑦 = 𝑒 3 2 3 𝑒
  • C 𝑥 = 2 3 𝑒 𝑒 3 , 𝑦 = 𝑒 3 + 2 3 𝑒
  • D 𝑥 = 2 3 𝑒 + 𝑒 3 , 𝑦 = 𝑒 3 2 3 𝑒
  • E 𝑥 = 2 3 𝑒 + 𝑒 3 , 𝑦 = 𝑒 3 + 2 3 𝑒

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?

  • Aslope of 𝐿 = 1 4 , slope of 𝐿 = 2
  • Bslope of 𝐿 = 1 4 , slope of 𝐿 = 2
  • Cslope of 𝐿 = 1 2 , slope of 𝐿 = 2
  • Dslope of 𝐿 = 1 2 , slope of 𝐿 = 2
  • Eslope of 𝐿 = 1 2 , slope of 𝐿 = 2

Q2:

The figures show the vector field 𝑦 , 𝑥 3 2 𝑦 , 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 ?

  • A 1 4 and 2
  • B 1 4 and 2
  • C 1 2 and 2
  • D 1 2 and 2
  • E 1 2 and 1 4

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

  • A 𝑥 = 4 5 𝑒 4 5 𝑒 , 𝑦 = 2 5 𝑒 + 8 5 𝑒
  • B 𝑥 = 4 5 𝑒 4 5 𝑒 , 𝑦 = 2 5 𝑒 + 8 5 𝑒
  • C 𝑥 = 4 5 𝑒 + 4 5 𝑒 , 𝑦 = 2 5 𝑒 8 5 𝑒
  • D 𝑥 = 4 5 𝑒 + 4 5 𝑒 , 𝑦 = 2 5 𝑒 8 5 𝑒
  • E 𝑥 = 3 5 𝑒 4 5 𝑒 , 𝑦 = 3 5 𝑒 + 8 5 𝑒

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

  • A 𝑥 = 2 5 𝑒 3 5 𝑒 , 𝑦 = 1 5 𝑒 + 6 5 𝑒
  • B 𝑥 = 2 5 𝑒 + 3 5 𝑒 , 𝑦 = 1 5 𝑒 6 5 𝑒
  • C 𝑥 = 2 5 𝑒 + 3 5 𝑒 , 𝑦 = 1 5 𝑒 6 5 𝑒
  • D 𝑥 = 1 5 𝑒 3 5 𝑒 , 𝑦 = 2 5 𝑒 6 5 𝑒
  • E 𝑥 = 2 5 𝑒 3 5 𝑒 , 𝑦 = 1 5 𝑒 + 6 5 𝑒

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

  • A 𝑥 = 4 5 𝑒 + 6 5 𝑒 , 𝑦 = 2 5 𝑒 1 2 5 𝑒
  • B 𝑥 = 4 5 𝑒 6 5 𝑒 , 𝑦 = 2 5 𝑒 + 1 2 5 𝑒
  • C 𝑥 = 3 5 𝑒 6 5 𝑒 , 𝑦 = 3 5 𝑒 1 2 5 𝑒
  • D 𝑥 = 4 5 𝑒 6 5 𝑒 , 𝑦 = 2 5 𝑒 + 1 2 5 𝑒
  • E 𝑥 = 4 5 𝑒 + 6 5 𝑒 , 𝑦 = 2 5 𝑒 1 2 5 𝑒

Using the fact that 𝑒 𝑒 = 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.

  • A 𝑥 + 1 2 𝑦 1 6 𝑥 + 1 3 𝑦 = 1
  • B 𝑥 + 1 6 𝑦 1 2 𝑥 + 1 3 𝑦 = 1
  • C 𝑥 + 1 3 𝑦 1 2 𝑥 1 6 𝑦 = 1
  • D 𝑥 + 1 2 𝑦 1 6 𝑥 1 3 𝑦 = 1
  • E 𝑥 + 1 6 𝑦 1 2 𝑥 1 3 𝑦 = 1

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?

  • Aslope of 𝐿 = 1 2 , slope of 𝐿 = 2
  • Bslope of 𝐿 = 1 2 , slope of 𝐿 = 1
  • Cslope of 𝐿 = 1 2 , slope of 𝐿 = 2
  • Dslope of 𝐿 = 1 4 , slope of 𝐿 = 1
  • Eslope of 𝐿 = 1 3 , slope of 𝐿 = 2

Q3:

Consider the parametric curve 𝑥 = 𝑒 ( 𝑏 𝑡 ) c o s , 𝑦 = 𝑒 ( 𝑏 𝑡 ) s i n with constants 𝑎 and 𝑏 . The figure shows the case 𝑎 = 1 5 and 𝑏 = 5 for 𝜋 𝑡 2 𝜋 .

Find a vector field such that the curve 𝑥 = 𝑒 ( 𝑏 𝑡 ) c o s and 𝑦 = 𝑒 ( 𝑏 𝑡 ) s i n is its integral curve.

  • A 𝑎 𝑥 𝑏 𝑦 , 𝑏 𝑥 + 𝑎 𝑦
  • B 𝑏 𝑥 𝑎 𝑦 , 𝑏 𝑥 + 𝑎 𝑦
  • C 𝑏 𝑥 + 𝑎 𝑦 , 𝑎 𝑥 𝑏 𝑦
  • D 𝑎 𝑥 𝑏 𝑦 , 𝑎 𝑥 + 𝑏 𝑦
  • E 𝑎 𝑥 + 𝑏 𝑦 , 𝑏 𝑥 𝑎 𝑦

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

  • A 𝑥 𝑎 𝑥 + 2 𝑎 + 𝑏 𝑥 = 0
  • B 𝑥 𝑎 𝑥 + 𝑎 + 2 𝑏 𝑥 = 0
  • C 𝑥 2 𝑎 𝑥 + 𝑎 + 𝑏 𝑥 = 0
  • D 𝑥 + 𝑎 𝑥 + 𝑎 𝑏 𝑥 = 0
  • E 2 𝑥 + 𝑎 𝑥 + 𝑎 𝑏 𝑥 = 0

You can check that 𝑥 = 𝑒 ( 𝑏 𝑡 ) s i n is also a solution to this differential equation and therefore any 𝑥 = 𝑓 ( 𝑡 ) = 𝑃 𝑒 ( 𝑏 𝑡 ) + 𝑄 𝑒 ( 𝑏 𝑡 ) c o s s i n 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.

  • A 𝑄 𝑒 ( 𝑏 𝑡 ) + 𝑃 𝑒 ( 𝑏 𝑡 ) c o s s i n
  • B 𝑄 𝑒 ( 𝑏 𝑡 ) + 𝑃 𝑒 ( 𝑏 𝑡 ) c o s s i n
  • C 𝑄 𝑒 ( 𝑎 𝑡 ) + 𝑃 𝑒 ( 𝑎 𝑡 ) c o s s i n
  • D 𝑄 𝑒 ( 𝑎 𝑡 ) + 𝑃 𝑒 ( 𝑎 𝑡 ) c o s s i n
  • E 𝑄 𝑒 ( 𝑏 𝑡 ) + 𝑃 𝑒 ( 𝑎 𝑡 ) c o s s i n

For the case 𝑎 = 1 5 and 𝑏 = 5 , find parametric equations for the integral curve that starts at the point ( 3 , 2 ) when 𝑡 = 0 .

  • A 𝑥 = 2 𝑒 ( 5 𝑡 ) 2 𝑒 ( 5 𝑡 ) c o s s i n , 𝑦 = 3 𝑒 ( 5 𝑡 ) + 3 𝑒 ( 5 𝑡 ) c o s s i n
  • B 𝑥 = 3 𝑒 ( 5 𝑡 ) 2 𝑒 ( 5 𝑡 ) c o s s i n , 𝑦 = 2 𝑒 ( 5 𝑡 ) + 3 𝑒 ( 5 𝑡 ) c o s s i n
  • C 𝑥 = 2 𝑒 ( 5 𝑡 ) + 2 𝑒 ( 5 𝑡 ) c o s s i n , 𝑦 = 3 𝑒 ( 5 𝑡 ) + 3 𝑒 ( 5 𝑡 ) c o s s i n
  • D 𝑥 = 𝑒 ( 5 𝑡 ) 2 𝑒 ( 5 𝑡 ) c o s s i n , 𝑦 = 3 𝑒 ( 5 𝑡 ) + 3 𝑒 ( 5 𝑡 ) c o s s i n
  • E 𝑥 = 3 𝑒 ( 5 𝑡 ) + 2 𝑒 ( 5 𝑡 ) c o s s i n , 𝑦 = 2 𝑒 ( 5 𝑡 ) 3 𝑒 ( 5 𝑡 ) c o s s i n

Q4:

The figures show the vector field 9 𝑦 , 𝑥 + 6 𝑦 , together with several of its flows.