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 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‘˜?

  • A13 and 2
  • B12 and 3
  • C13 and 3
  • D12 and 2
  • E14 and 2

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

  • Aπ‘₯=βˆ’43𝑒+13𝑒,𝑦=βˆ’23𝑒+23π‘’ο‘‰οŽ‘ο‘‰οŽ‘οŠ¨οοŠ¨ο
  • Bπ‘₯=βˆ’43𝑒+13𝑒,𝑦=βˆ’23𝑒+23π‘’ο‘‰οŽ’ο‘‰οŽ‘οŠ¨οοŠ¨ο
  • Cπ‘₯=43𝑒+13𝑒,𝑦=23π‘’βˆ’23π‘’ο‘‰οŽ’ο‘‰οŽ‘οŠ¨οοŠ¨ο
  • Dπ‘₯=βˆ’43𝑒+13𝑒,𝑦=23π‘’βˆ’23π‘’ο‘‰οŽ‘ο‘‰οŽ‘οŠ¨οοŠ¨ο
  • Eπ‘₯=43π‘’βˆ’13𝑒,𝑦=βˆ’23𝑒+23π‘’ο‘‰οŽ‘ο‘‰οŽ‘οŠ¨οοŠ¨ο

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

  • Aπ‘₯=43π‘’βˆ’43𝑒,𝑦=23π‘’βˆ’83π‘’ο‘‰οŽ’ο‘‰οŽ‘οŠ¨οοŠ¨ο
  • Bπ‘₯=43𝑒+43𝑒,𝑦=23𝑒+83π‘’ο‘‰οŽ‘ο‘‰οŽ‘οŠ¨οοŠ¨ο
  • Cπ‘₯=βˆ’43π‘’βˆ’43𝑒,𝑦=23π‘’βˆ’83π‘’ο‘‰οŽ‘ο‘‰οŽ‘οŠ¨οοŠ¨ο
  • Dπ‘₯=43π‘’βˆ’43𝑒,𝑦=βˆ’23𝑒+83π‘’ο‘‰οŽ‘ο‘‰οŽ‘οŠ¨οοŠ¨ο
  • Eπ‘₯=43𝑒+43𝑒,𝑦=βˆ’23𝑒+83π‘’ο‘‰οŽ’ο‘‰οŽ‘οŠ¨οοŠ¨ο

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

  • Aπ‘₯=βˆ’23π‘’βˆ’π‘’3,𝑦=𝑒3βˆ’23π‘’ο‘‰οŽ’ο‘‰οŽ‘οŠ¨οοŠ¨ο
  • Bπ‘₯=23𝑒+𝑒3,𝑦=βˆ’π‘’3+23π‘’ο‘‰οŽ‘ο‘‰οŽ‘οŠ¨οοŠ¨ο
  • Cπ‘₯=βˆ’23π‘’βˆ’π‘’3,𝑦=𝑒3+23π‘’ο‘‰οŽ‘ο‘‰οŽ‘οŠ¨οοŠ¨ο
  • Dπ‘₯=βˆ’23𝑒+𝑒3,𝑦=𝑒3βˆ’23π‘’ο‘‰οŽ‘ο‘‰οŽ‘οŠ¨οοŠ¨ο
  • Eπ‘₯=βˆ’23𝑒+𝑒3,𝑦=βˆ’π‘’3+23π‘’ο‘‰οŽ’ο‘‰οŽ‘οŠ¨οοŠ¨ο

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?

  • Aslope of 𝐿=βˆ’12, slope of 𝐿=βˆ’2
  • Bslope of 𝐿=βˆ’12, slope of 𝐿=2
  • Cslope of 𝐿=12, slope of 𝐿=2
  • Dslope of 𝐿=14, slope of 𝐿=2
  • Eslope of 𝐿=βˆ’14, slope of 𝐿=βˆ’2

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‘˜?

  • A14 and 2
  • B14 and βˆ’2
  • C12 and βˆ’2
  • D12 and 2
  • E12 and 14

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

  • Aπ‘₯=35π‘’βˆ’45π‘’ο‘‰οŽ‘οŠ±οŠ¨ο, 𝑦=35𝑒+85π‘’ο‘‰οŽ‘οŠ±οŠ¨ο
  • Bπ‘₯=45π‘’βˆ’45π‘’ο‘‰οŽ‘οŠ±οŠ¨ο, 𝑦=25𝑒+85π‘’ο‘‰οŽ‘οŠ±οŠ¨ο
  • Cπ‘₯=45𝑒+45π‘’ο‘‰οŽ‘οŠ±οŠ¨ο, 𝑦=25π‘’βˆ’85π‘’ο‘‰οŽ‘οŠ±οŠ¨ο
  • Dπ‘₯=45𝑒+45π‘’ο‘‰οŽ‘οŠ¨ο, 𝑦=25π‘’βˆ’85π‘’ο‘‰οŽ‘οŠ¨ο
  • Eπ‘₯=45π‘’βˆ’45π‘’ο‘‰οŽ‘οŠ¨ο, 𝑦=25𝑒+85π‘’ο‘‰οŽ‘οŠ¨ο

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

  • Aπ‘₯=βˆ’25𝑒+35π‘’ο‘‰οŽ‘οŠ±οŠ¨ο, 𝑦=βˆ’15π‘’βˆ’65π‘’ο‘‰οŽ‘οŠ±οŠ¨ο
  • Bπ‘₯=βˆ’15π‘’βˆ’35π‘’ο‘‰οŽ‘οŠ±οŠ¨ο, 𝑦=βˆ’25π‘’βˆ’65π‘’ο‘‰οŽ‘οŠ±οŠ¨ο
  • Cπ‘₯=βˆ’25π‘’βˆ’35π‘’ο‘‰οŽ‘οŠ¨ο, 𝑦=βˆ’15𝑒+65π‘’ο‘‰οŽ‘οŠ¨ο
  • Dπ‘₯=βˆ’25π‘’βˆ’35π‘’ο‘‰οŽ‘οŠ±οŠ¨ο, 𝑦=βˆ’15𝑒+65π‘’ο‘‰οŽ‘οŠ±οŠ¨ο
  • Eπ‘₯=βˆ’25𝑒+35π‘’ο‘‰οŽ‘οŠ¨ο, 𝑦=βˆ’15π‘’βˆ’65π‘’ο‘‰οŽ‘οŠ¨ο

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

  • Aπ‘₯=45𝑒+65π‘’ο‘‰οŽ‘οŠ¨ο, 𝑦=25π‘’βˆ’125π‘’ο‘‰οŽ‘οŠ¨ο
  • Bπ‘₯=45π‘’βˆ’65π‘’ο‘‰οŽ‘οŠ±οŠ¨ο, 𝑦=25𝑒+125π‘’ο‘‰οŽ‘οŠ±οŠ¨ο
  • Cπ‘₯=45π‘’βˆ’65π‘’ο‘‰οŽ‘οŠ¨ο, 𝑦=25𝑒+125π‘’ο‘‰οŽ‘οŠ¨ο
  • Dπ‘₯=45𝑒+65π‘’ο‘‰οŽ‘οŠ±οŠ¨ο, 𝑦=25π‘’βˆ’125π‘’ο‘‰οŽ‘οŠ±οŠ¨ο
  • Eπ‘₯=35π‘’βˆ’65π‘’ο‘‰οŽ‘οŠ±οŠ¨ο, 𝑦=35π‘’βˆ’125π‘’ο‘‰οŽ‘οŠ±οŠ¨ο

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.

  • Aο€Όπ‘₯+12π‘¦οˆο€Ό16π‘₯+13π‘¦οˆ=1οŠͺ
  • Bο€Όπ‘₯+12π‘¦οˆο€Ό16π‘₯βˆ’13π‘¦οˆ=1οŠͺ
  • Cο€Όπ‘₯+16π‘¦οˆο€Ό12π‘₯+13π‘¦οˆ=1οŠͺ
  • Dο€Όπ‘₯+16π‘¦οˆο€Ό12π‘₯βˆ’13π‘¦οˆ=1οŠͺ
  • Eο€Όπ‘₯+13π‘¦οˆο€Ό12π‘₯βˆ’16π‘¦οˆ=1οŠͺ

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?

  • ASlope of 𝐿=12, slope of 𝐿=βˆ’2
  • BSlope of 𝐿=14, slope of 𝐿=βˆ’1
  • CSlope of 𝐿=13, slope of 𝐿=βˆ’2
  • DSlope of 𝐿=12, slope of 𝐿=2
  • ESlope of 𝐿=12, slope of 𝐿=1

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.

  • AβŸ¨π‘π‘₯βˆ’π‘Žπ‘¦,𝑏π‘₯+π‘Žπ‘¦βŸ©
  • BβŸ¨π‘Žπ‘₯+𝑏𝑦,𝑏π‘₯βˆ’π‘Žπ‘¦βŸ©
  • CβŸ¨π‘Žπ‘₯βˆ’π‘π‘¦,𝑏π‘₯+π‘Žπ‘¦βŸ©
  • DβŸ¨π‘Žπ‘₯βˆ’π‘π‘¦,π‘Žπ‘₯+π‘π‘¦βŸ©
  • EβŸ¨π‘π‘₯+π‘Žπ‘¦,π‘Žπ‘₯βˆ’π‘π‘¦βŸ©

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

  • Aπ‘₯β€²β€²βˆ’2π‘Žπ‘₯β€²+ο€Ήπ‘Ž+𝑏π‘₯=0
  • B2π‘₯β€²β€²+π‘Žπ‘₯β€²+ο€Ήπ‘Žβˆ’π‘ο…π‘₯=0
  • Cπ‘₯β€²β€²βˆ’π‘Žπ‘₯β€²+ο€Ή2π‘Ž+𝑏π‘₯=0
  • Dπ‘₯β€²β€²βˆ’π‘Žπ‘₯β€²+ο€Ήπ‘Ž+2𝑏π‘₯=0
  • Eπ‘₯β€²β€²+π‘Žπ‘₯β€²+ο€Ήπ‘Žβˆ’π‘ο…π‘₯=0

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.

  • Aβˆ’π‘„π‘’(𝑏𝑑)+𝑃𝑒(𝑏𝑑)cossin
  • Bβˆ’π‘„π‘’(𝑏𝑑)+𝑃𝑒(π‘Žπ‘‘)cossin
  • C𝑄𝑒(𝑏𝑑)+𝑃𝑒(𝑏𝑑)cossin
  • Dβˆ’π‘„π‘’(π‘Žπ‘‘)+𝑃𝑒(π‘Žπ‘‘)cossin
  • E𝑄𝑒(π‘Žπ‘‘)+𝑃𝑒(π‘Žπ‘‘)cossin

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.

  • Aπ‘₯=2𝑒(5𝑑)βˆ’2𝑒(5𝑑)οŠ¨ο‘‰οŽ€ο‘‰οŽ€cossin, 𝑦=3𝑒(5𝑑)+3𝑒(5𝑑)οŠ¨ο‘‰οŽ€ο‘‰οŽ€cossin
  • Bπ‘₯=3𝑒(5𝑑)+2𝑒(5𝑑)ο‘‰οŽ€ο‘‰οŽ€cossin, 𝑦=2𝑒(5𝑑)βˆ’3𝑒(5𝑑)ο‘‰οŽ€ο‘‰οŽ€cossin
  • Cπ‘₯=3𝑒(5𝑑)βˆ’2𝑒(5𝑑)ο‘‰οŽ€ο‘‰οŽ€cossin, 𝑦=2𝑒(5𝑑)+3𝑒(5𝑑)ο‘‰οŽ€ο‘‰οŽ€cossin
  • Dπ‘₯=𝑒(5𝑑)βˆ’2𝑒(5𝑑)ο‘‰οŽ€ο‘‰οŽ€cossin, 𝑦=3𝑒(5𝑑)+3𝑒(5𝑑)ο‘‰οŽ€ο‘‰οŽ€cossin
  • Eπ‘₯=2𝑒(5𝑑)+2𝑒(5𝑑)οŠ¨ο‘‰οŽ€ο‘‰οŽ€cossin, 𝑦=3𝑒(5𝑑)+3𝑒(5𝑑)ο‘‰οŽ€ο‘‰οŽ€cossin

Q4:

The figures show the vector field βŸ¨βˆ’9𝑦,π‘₯+6π‘¦βŸ©, 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‘˜?

In this case, where π‘˜ is a repeated root, linear combinations of 𝑑𝑒 and 𝑒 are used. Hence, find the parametric equations of the integral curve that is at (0,2) when 𝑑=0.

  • Aπ‘₯=βˆ’9π‘‘π‘’οŠ©ο, 𝑦=2𝑒+3π‘‘π‘’οŠ©οοŠ©ο
  • Bπ‘₯=βˆ’9π‘’οŠ©ο, 𝑦=2π‘’βˆ’3π‘‘π‘’οŠ©οοŠ©ο
  • Cπ‘₯=βˆ’3π‘‘π‘’οŠ©ο, 𝑦=2𝑒+2π‘‘π‘’οŠ©οοŠ©ο
  • Dπ‘₯=βˆ’18π‘‘π‘’οŠ©ο, 𝑦=2𝑒+6π‘‘π‘’οŠ©οοŠ©ο
  • Eπ‘₯=18π‘‘π‘’οŠ©ο, 𝑦=2π‘’βˆ’6π‘‘π‘’οŠ©οοŠ©ο

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

  • Aπ‘₯=βˆ’π‘’βˆ’2π‘‘π‘’οŠ©οοŠ©ο, 𝑦=𝑒+π‘‘π‘’οŠ©οοŠ©ο
  • Bπ‘₯=π‘’βˆ’6π‘‘π‘’οŠ©οοŠ©ο, 𝑦=π‘’βˆ’2π‘‘π‘’οŠ©οοŠ©ο
  • Cπ‘₯=βˆ’π‘’βˆ’6π‘‘π‘’οŠ©οοŠ©ο, 𝑦=𝑒+2π‘‘π‘’οŠ©οοŠ©ο
  • Dπ‘₯=βˆ’π‘’+6π‘‘π‘’οŠ¨οοŠ¨ο, 𝑦=π‘’βˆ’2π‘‘π‘’οŠ¨οοŠ¨ο
  • Eπ‘₯=βˆ’π‘’βˆ’6π‘‘π‘’οŠ¨οοŠ¨ο, 𝑦=𝑒+2π‘‘π‘’οŠ¨οοŠ¨ο

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

  • Aπ‘₯=βˆ’π‘’+3π‘‘π‘’οŠ©οοŠ©ο, 𝑦=βˆ’π‘‘π‘’οŠ©ο
  • Bπ‘₯=βˆ’π‘’βˆ’3π‘‘π‘’οŠ©οοŠ©ο, 𝑦=π‘‘π‘’οŠ©ο
  • Cπ‘₯=βˆ’π‘’βˆ’2π‘‘π‘’οŠ©οοŠ©ο, 𝑦=βˆ’π‘‘π‘’οŠ©ο
  • Dπ‘₯=βˆ’π‘’βˆ’π‘’οŠ©οοŠ©ο, 𝑦=π‘‘π‘’οŠ©ο
  • Eπ‘₯=βˆ’π‘’βˆ’3π‘‘π‘’οŠ©οοŠ©ο, 𝑦=π‘‘π‘’οŠ©ο

As π‘‘β†’βˆž and π‘‘β†’βˆ’βˆž along an integral curve, the secant between (0, 0) and (𝑓(𝑑),𝑔(𝑑)) approaches the dashed line shown. What is the slope of this line?

  • Aβˆ’13
  • B13
  • C12
  • Dβˆ’12
  • Eβˆ’14

Q5:

If π‘₯=𝑓(𝑑) parameterizes an integral curve of the vector field 𝑉(π‘₯,𝑦)=βŸ¨π‘¦,π‘₯⟩, then 𝑓′′=𝑓. This means 𝑓 is a linear combination of 𝑒 and π‘’οŠ±ο.

Find the π‘₯-parameter function 𝑓(𝑑) for the integral curve to this vector field that starts at the point (2,3).

  • A𝑓(𝑑)=52π‘’βˆ’12π‘’οοŠ±ο
  • B𝑓(𝑑)=12π‘’βˆ’52π‘’οοŠ±ο
  • C𝑓(𝑑)=32π‘’βˆ’23π‘’οοŠ±ο
  • D𝑓(𝑑)=52𝑒+12π‘’οοŠ±ο
  • E𝑓(𝑑)=12𝑒+52π‘’οοŠ±ο

Find the Cartesian equation of the integral curve determined above.

Hint: It is a hyperbola.

  • Aπ‘₯βˆ’π‘¦=βˆ’5
  • Bπ‘₯+𝑦=βˆ’5
  • Cπ‘₯+𝑦=5
  • Dπ‘₯βˆ’2𝑦=βˆ’5
  • Eπ‘₯βˆ’π‘¦=5

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

  • A𝑦=π‘₯
  • B𝑦=βˆ’π‘₯
  • C𝑦=π‘₯
  • D𝑦=βˆ’π‘₯
  • E𝑦=π‘₯+2

Q6:

An integral curve (or flow) of a vector field 𝑉 is a parametric curve π‘₯=𝑓(𝑑),𝑦=𝑔(𝑑) with βŸ¨π‘“(𝑑),𝑔(𝑑)⟩=𝑉(𝑓(𝑑),𝑔(𝑑)) for every 𝑑 where 𝑓 and 𝑔 are defined.

By solving the equations 𝑓(𝑑)=1 and 𝑔(𝑑)=2, find an integral curve for the vector field 𝑉(π‘₯,𝑦)=⟨1,2⟩ that also satisfies (𝑓(0),𝑔(0))=(βˆ’1,1).

  • A𝑓(𝑑)=2π‘‘βˆ’1𝑔(𝑑)=2𝑑+1,
  • B𝑓(𝑑)=π‘‘βˆ’1,𝑔(𝑑)=2π‘‘βˆ’1
  • C𝑓(𝑑)=π‘‘βˆ’1,𝑔(𝑑)=2𝑑+1
  • D𝑓(𝑑)=2𝑑+1𝑔(𝑑)=π‘‘βˆ’1,
  • E𝑓(𝑑)=𝑑+1,𝑔(𝑑)=2𝑑+1

Consider the vector field 𝑉(π‘₯,𝑦)=⟨1,2⟩. Find the Cartesian equation of the vector field’s integral curve which is at the point (2,βˆ’3) when 𝑑=0.

  • A𝑦+2π‘₯=βˆ’7
  • Bπ‘¦βˆ’π‘₯=βˆ’7
  • Cπ‘¦βˆ’2π‘₯=βˆ’7
  • Dπ‘¦βˆ’2π‘₯=7
  • E2π‘¦βˆ’π‘₯=βˆ’7

Find the Cartesian equation of the integral curve to 𝑉(π‘₯,𝑦)=⟨1,π‘₯⟩ that starts at the point (2, 2).

  • A𝑦=π‘₯3
  • B𝑦=π‘₯βˆ’5π‘₯+102
  • C𝑦=π‘₯βˆ’4π‘₯+82
  • D𝑦=π‘₯βˆ’2
  • E𝑦=π‘₯2

Find the Cartesian equation of the integral curve to 𝑉(π‘₯,𝑦)=⟨π‘₯,π‘₯⟩ that starts at the point (2, 2).

  • A𝑦=π‘₯2
  • B𝑦=(π‘₯βˆ’1)2+2(π‘₯βˆ’1)+(π‘₯βˆ’1)βˆ’12ln
  • C𝑦=π‘₯βˆ’4π‘₯+82
  • D𝑦=π‘₯βˆ’2
  • E𝑦=π‘₯3

Find the parametric equations of the integral curve to 𝑉(π‘₯,𝑦)=⟨π‘₯,π‘₯⟩ that starts at the point (0, 2).

  • A𝑓(𝑑)=1,𝑔(𝑑)=0
  • B𝑓(𝑑)=βˆ’1,𝑔(𝑑)=3
  • C𝑓(𝑑)=0,𝑔(𝑑)=2
  • D𝑓(𝑑)=βˆ’1,𝑔(𝑑)=2
  • E𝑓(𝑑)=0,𝑔(𝑑)=3

Do the integral curves of the vector fields ⟨1,π‘₯⟩ and ⟨π‘₯,π‘₯⟩ starting at (0, 2) describe the same set in β„οŠ¨ for 𝑑β‰₯0?

  • Ayes
  • Bno

Do the integral curves of the vector fields ⟨1,π‘₯⟩ and ⟨π‘₯,π‘₯⟩ starting at (2,2) describe the same set in β„οŠ¨ for 𝑑β‰₯0?

  • Ayes
  • Bno

The integral curves to ⟨1,π‘₯⟩ and ⟨π‘₯,π‘₯⟩ that start at (βˆ’2,4) both lie inside the curve 𝑦=π‘₯2+2 but go in opposite directions. Determine the parametric equations that integrate the vector field ⟨π‘₯,π‘₯⟩ and start at (βˆ’2,4).

  • A𝑓(𝑑)=βˆ’32𝑑+3,𝑔(𝑑)=2(2𝑑+1)+2
  • B𝑓(𝑑)=βˆ’22𝑑+1,𝑔(𝑑)=2(2𝑑+1)+2
  • C𝑓(𝑑)=βˆ’22𝑑+1,𝑔(𝑑)=3(2𝑑+3)+5
  • D𝑓(𝑑)=βˆ’2𝑑+1,𝑔(𝑑)=2(𝑑+1)+5

Do the integral curves of the vector fields ⟨1,π‘₯⟩ and ⟨π‘₯,π‘₯⟩ starting at (0, 2) describe the same set in β„οŠ¨ for 𝑑β‰₯0?

  • Ano
  • Byes

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