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

Q1:

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.

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

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.

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 .

Q2:

The figures show the vector field ⟨ − 9 𝑦 , 𝑥 + 6 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 𝑘 𝑡 . What are the values of 𝑘 ?

In this case, where 𝑘 is a repeated root, linear combinations of 𝑡 𝑒 𝑘 𝑡 and 𝑒 𝑘 𝑡 are used. Hence, find 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 ( − 1 , 0 ) when 𝑡 = 0 ?

As 𝑡 → ∞ and 𝑡 → − ∞ along an integral curve, the secant between (0, 0) and ( 𝑓 ( 𝑡 ) , 𝑔 ( 𝑡 ) ) approaches the dashed line shown. What is the slope of this line?

Q3:

If 𝑥 = 𝑓 ( 𝑡 ) parameterizes an integral curve of the vector field 𝑉 ( 𝑥 , 𝑦 ) = ⟨ 𝑦 , 𝑥 ⟩ , then 𝑓 ′ ′ = 𝑓 . This means 𝑓 is a linear combination of 𝑒 and 𝑒 .

Find the 𝑥 -parameter function 𝑓 ( 𝑡 ) for the integral curve to this vector field that starts at the point ( 2 , 3 ) .

Find the Cartesian equation of the integral curve determined above.

Hint: It is a hyperbola.

Find the Cartesian equation of the integral curve to this vector field that starts at the point ( 2 , 2 ) .

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