12-267/Homework Assignment 2 - Revision history

Twine: Added solution to task 2

2012-10-24T22:44:46Z

Added solution to task 2

← Older revision		Revision as of 18:44, 24 October 2012
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	{{Template:12-267:Dror/Students Divider}}		{{Template:12-267:Dror/Students Divider}}

			'''Solution to Task 2''' --[[User:Twine\|Twine]] 18:44, 24 October 2012 (EDT)

			'''Part 1'''

			What we want is an integration factor <math>\mu</math> such that <math>(\mu M)_y = (\mu N)_x</math>. Let <math>xy = z</math>

			When <math>\mu</math> is a function of z, <math>\frac{dv}{dx} = \frac{dv}{dz} \frac{dz}{dx} = \frac{dv}{dz} y</math>.

			Similarly <math>\frac{dv}{dy} = \frac{dv}{dz} x</math>.

			Then we have

			<math>(\mu M)_y = (\mu N)_x</math>

			<math>\iff \mu_{z}xM - \mu_{xy}yN = \mu (N_x - M_y)</math>

			<math>\iff \frac{\mu_{z}}{mu} = \frac{N_x - M_y}{xM - yN}</math>

			If the right hand side depends only on xy, we can get

			<math>\mu (xy) = e^{\int \frac{N_x - M_y}{xM - yN} d(xy)}</math>

			which satisfies the requirements of an integrating factor.

			'''Part 2'''

			As in Part 1, we need <math>(\mu M)_y = (\mu N)_x</math>. Let <math>z = x + y</math>

			When <math>\mu</math> is a function of <math>x + y</math>, <math>\frac{d \mu}{dx} = \frac{d\mu}{dz} \frac{dz}{dx} = \frac{d\mu}{dz}</math>.

			Similarly <math>\frac{d \mu}{dy} = \frac{d\mu}{dz}</math>

			Then we have

			<math>(\mu M)_y = (\mu N)_x</math>

			<math>\iff \frac{\mu_{z}}{\mu} = \frac{N_x - M_y}{M-N}</math>

			If the right hand side of this equation depends only on z (that is, only on (x+y)), then we have

			<math>\mu (x + y) = e^{\int \frac{N_x - M_y}{M - N} d(x + y)}</math>

			which satisfies the requirements of an integrating factor.

	Solution to HW2: [[User:Mathstudent\|Mathstudent]]		Solution to HW2: [[User:Mathstudent\|Mathstudent]]

Twine: Added student divider

2012-10-24T21:05:39Z

Added student divider

← Older revision		Revision as of 17:05, 24 October 2012
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	'''Task 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.		'''Task 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.

			{{Template:12-267:Dror/Students Divider}}

	Solution to HW2: [[User:Mathstudent\|Mathstudent]]		Solution to HW2: [[User:Mathstudent\|Mathstudent]]

Twine: Changed format of Mathstudent's solutions to a gallery

2012-10-24T20:58:13Z

Changed format of Mathstudent's solutions to a gallery

← Older revision		Revision as of 16:58, 24 October 2012
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	'''Task 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.		'''Task 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.

	~~[http://drorbn.net/index.php?title=Image:005.jpg~~ Solution to HW2~~, page 1]~~ [[User:Mathstudent\|Mathstudent]]		Solution to HW2: [[User:Mathstudent\|Mathstudent]]
			<gallery>
	[http://drorbn.net/index.php?title=Image:006.jpg Solution to HW2, page 2] [[User:Mathstudent\|Mathstudent]]
			Image:005.jpg\|page 1
	[http://drorbn.net/index.php?title=Image:007.jpg Solution to HW2, page 3] [[User:Mathstudent\|Mathstudent]]
			Image:006.jpg\|page 2
	[http://drorbn.net/index.php?title=Image:008.jpg Solution to HW2, page 4] [[User:Mathstudent\|Mathstudent]]
			Image:007.jpg\|page 3
	[http://drorbn.net/index.php?title=Image:009.jpg Solution to HW2, page 5] [[User:Mathstudent\|Mathstudent]]
			Image:008.jpg\|page 4
	[http://drorbn.net/index.php?title=Image:010.jpg Solution to HW2, page 6] [[User:Mathstudent\|Mathstudent]]
			Image:009.jpg\|page 5
	[http://drorbn.net/index.php?title=Image:011.jpg Solution to HW2, page 7] [[User:Mathstudent\|Mathstudent]]
			Image:010.jpg\|page 6
			Image:011.jpg\|page 7
			</gallery>

Mathstudent at 23:36, 23 October 2012

2012-10-23T23:36:43Z

← Older revision		Revision as of 19:36, 23 October 2012
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	[http://drorbn.net/index.php?title=Image:009.jpg Solution to HW2, page 5] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:009.jpg Solution to HW2, page 5] [[User:Mathstudent\|Mathstudent]]
	[http://drorbn.net/index.php?title=Image:010.jpg Solution to HW2, page 6] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:010.jpg Solution to HW2, page 6] [[User:Mathstudent\|Mathstudent]]
	[http://drorbn.net/index.php?title=Image:011.jpg Solution to HW2, page 6] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:011.jpg Solution to HW2, page 7] [[User:Mathstudent\|Mathstudent]]

Mathstudent at 23:26, 23 October 2012

2012-10-23T23:26:37Z

← Older revision		Revision as of 19:26, 23 October 2012
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	'''Task 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.		'''Task 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.

			[http://drorbn.net/index.php?title=Image:005.jpg Solution to HW2, page 1] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:006.jpg Solution to HW2, page 2] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:007.jpg Solution to HW2, page 3] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:008.jpg Solution to HW2, page 4] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:009.jpg Solution to HW2, page 5] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:010.jpg Solution to HW2, page 6] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:011.jpg Solution to HW2, page 6] [[User:Mathstudent\|Mathstudent]]

Drorbn at 20:00, 25 September 2012

2012-09-25T20:00:41Z

Drorbn at 18:20, 24 September 2012

2012-09-24T18:20:08Z

← Older revision		Revision as of 14:20, 24 September 2012
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	{{12-267/Navigation}}		{{12-267/Navigation}}
	{{In Preparation}}

	This assignment is due at the tutorial on Tuesday October 2. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the your readers will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due at the tutorial on Tuesday October 2. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the your readers will be having hard time deciphering your work they will give up and assume it is wrong.

Drorbn at 18:15, 24 September 2012

2012-09-24T18:15:09Z

← Older revision		Revision as of 14:15, 24 September 2012
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	# <math>\frac{dy}{dx}=\frac{2y-x+5}{2x-y-4}</math> (hint: consider trying <math>x_1=x+\alpha</math> and <math>y_1=y+\beta</math> for good <math>\alpha</math>, <math>\beta</math>).		# <math>\frac{dy}{dx}=\frac{2y-x+5}{2x-y-4}</math> (hint: consider trying <math>x_1=x+\alpha</math> and <math>y_1=y+\beta</math> for good <math>\alpha</math>, <math>\beta</math>).
	# <math>y'=\frac{y^3}{1-2xy^2}</math> with <math>y(0)=1</math>.		# <math>y'=\frac{y^3}{1-2xy^2}</math> with <math>y(0)=1</math>.
	# <math>y'=\frac{x^2-1}{y^2+1}</math> with <math>y(-1)=1</math>.
	# <math>\frac{dy}{dx}=\frac{2y+\sqrt{x^2-y^2}}{2x}</math>.		# <math>\frac{dy}{dx}=\frac{2y+\sqrt{x^2-y^2}}{2x}</math>.
	# <math>0=(\cos 2y - \sin x)dx-2\tan x\sin 2y dy</math>.

	'''Question 2.''' Let <math>M</math> and <math>N</math> be differentiable functions of <math>x</math> and <math>y</math>.		'''Question 2.''' Let <math>M</math> and <math>N</math> be differentiable functions of <math>x</math> and <math>y</math>.

Drorbn at 17:03, 24 September 2012

2012-09-24T17:03:50Z

← Older revision		Revision as of 13:03, 24 September 2012
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	# <math>y'=\frac{x^2-1}{y^2+1}</math> with <math>y(-1)=1</math>.		# <math>y'=\frac{x^2-1}{y^2+1}</math> with <math>y(-1)=1</math>.
	# <math>\frac{dy}{dx}=\frac{2y+\sqrt{x^2-y^2}}{2x}</math>.		# <math>\frac{dy}{dx}=\frac{2y+\sqrt{x^2-y^2}}{2x}</math>.
	# <math>0=(\cos 2y - \sin x)dx-2\tan x\sin 2y</math>.		# <math>0=(\cos 2y - \sin x)dx-2\tan x\sin 2y dy</math>.

	'''Question 2.''' Let <math>M</math> and <math>N</math> be differentiable functions of <math>x</math> and <math>y</math>.		'''Question 2.''' Let <math>M</math> and <math>N</math> be differentiable functions of <math>x</math> and <math>y</math>.
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	'''Question 3.''' The equation <math>y'+p(x)y=q(x)y^n</math> is called a "Bernoulli Equation".		'''Question 3.''' The equation <math>y'+p(x)y=q(x)y^n</math> is called a "Bernoulli Equation".
	# Explain why you already know how to solve the Bernoulli equation when <math>n=0</math> and when <math>n=1</math>.		# Explain why you already know how to solve the Bernoulli equation when <math>n=0</math> and when <math>n=1</math>.
	# Show that if n\neq0,1, then the substitution <math>v=y^{1-n}</math> reduces the Bernoulli equation to an equation you already know how to solve.		# Show that if <math>n\neq0,1</math>, then the substitution <math>v=y^{1-n}</math> reduces the Bernoulli equation to an equation you already know how to solve.
	# Solve the equation <math>x^2y'+2xy-y^3=0</math> (for <math>x>0</math>).		# Solve the equation <math>x^2y'+2xy-y^3=0</math> (for <math>x>0</math>).

Drorbn at 16:58, 24 September 2012

2012-09-24T16:58:32Z

← Older revision		Revision as of 12:58, 24 September 2012
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	# <math>\frac{dy}{dx}=\frac{2y+\sqrt{x^2-y^2}}{2x}</math>.		# <math>\frac{dy}{dx}=\frac{2y+\sqrt{x^2-y^2}}{2x}</math>.
	# <math>0=(\cos 2y - \sin x)dx-2\tan x\sin 2y</math>.		# <math>0=(\cos 2y - \sin x)dx-2\tan x\sin 2y</math>.

			'''Question 2.''' Let <math>M</math> and <math>N</math> be differentiable functions of <math>x</math> and <math>y</math>.
			# Show that if <math>\frac{N_x-M_y}{xM-yN}</math> depends only on <math>xy</math>, then the differential equation <math>Mdx+Ndy=0</math> has an integrating factor of the form <math>\mu(xy)</math>, where <math>\mu</math> is a function of a single variable.
			# Find a condition on <math>M</math> and <math>N</math> that would imply that the differential equation <math>Mdx+Ndy=0</math> would have an integrating factor of the form <math>\mu(x+y)</math>, where <math>\mu</math> is a function of a single variable.

			'''Question 3.''' The equation <math>y'+p(x)y=q(x)y^n</math> is called a "Bernoulli Equation".
			# Explain why you already know how to solve the Bernoulli equation when <math>n=0</math> and when <math>n=1</math>.
			# Show that if n\neq0,1, then the substitution <math>v=y^{1-n}</math> reduces the Bernoulli equation to an equation you already know how to solve.
			# Solve the equation <math>x^2y'+2xy-y^3=0</math> (for <math>x>0</math>).

			'''Question 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.

← Older revision		Revision as of 16:58, 24 October 2012
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	'''Task 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.		'''Task 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.

	~~[http://drorbn.net/index.php?title=Image:005.jpg~~ Solution to HW2~~, page 1]~~ [[User:Mathstudent\|Mathstudent]]		Solution to HW2: [[User:Mathstudent\|Mathstudent]]
			<gallery>
	[http://drorbn.net/index.php?title=Image:006.jpg Solution to HW2, page 2] [[User:Mathstudent\|Mathstudent]]
			Image:005.jpg\|page 1
	[http://drorbn.net/index.php?title=Image:007.jpg Solution to HW2, page 3] [[User:Mathstudent\|Mathstudent]]
			Image:006.jpg\|page 2
	[http://drorbn.net/index.php?title=Image:008.jpg Solution to HW2, page 4] [[User:Mathstudent\|Mathstudent]]
			Image:007.jpg\|page 3
	[http://drorbn.net/index.php?title=Image:009.jpg Solution to HW2, page 5] [[User:Mathstudent\|Mathstudent]]
			Image:008.jpg\|page 4
	[http://drorbn.net/index.php?title=Image:010.jpg Solution to HW2, page 6] [[User:Mathstudent\|Mathstudent]]
			Image:009.jpg\|page 5
	[http://drorbn.net/index.php?title=Image:011.jpg Solution to HW2, page 7] [[User:Mathstudent\|Mathstudent]]
			Image:010.jpg\|page 6
			Image:011.jpg\|page 7
			</gallery>

← Older revision		Revision as of 19:36, 23 October 2012
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	[http://drorbn.net/index.php?title=Image:009.jpg Solution to HW2, page 5] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:009.jpg Solution to HW2, page 5] [[User:Mathstudent\|Mathstudent]]
	[http://drorbn.net/index.php?title=Image:010.jpg Solution to HW2, page 6] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:010.jpg Solution to HW2, page 6] [[User:Mathstudent\|Mathstudent]]
	[http://drorbn.net/index.php?title=Image:011.jpg Solution to HW2, page 6] [[User:Mathstudent\|Mathstudent]]		[http://drorbn.net/index.php?title=Image:011.jpg Solution to HW2, page 7] [[User:Mathstudent\|Mathstudent]]

← Older revision		Revision as of 19:26, 23 October 2012
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	'''Task 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.		'''Task 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.

			[http://drorbn.net/index.php?title=Image:005.jpg Solution to HW2, page 1] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:006.jpg Solution to HW2, page 2] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:007.jpg Solution to HW2, page 3] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:008.jpg Solution to HW2, page 4] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:009.jpg Solution to HW2, page 5] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:010.jpg Solution to HW2, page 6] [[User:Mathstudent\|Mathstudent]]
			[http://drorbn.net/index.php?title=Image:011.jpg Solution to HW2, page 6] [[User:Mathstudent\|Mathstudent]]

← Older revision		Revision as of 16:00, 25 September 2012
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	This assignment is due at the tutorial on Tuesday October 2. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the your readers will be having hard time deciphering your work they will give up and assume it is wrong.		This assignment is due at the tutorial on Tuesday October 2. Here and everywhere, '''neatness counts!!''' You may be brilliant and you may mean just the right things, but if the your readers will be having hard time deciphering your work they will give up and assume it is wrong.

			'''Task 0.''' Identify yourself in the [[12-267/Class Photo\|Class Photo]]!
⚫	'''~~Question~~ 1.''' Solve the following differential equations:

		⚫	'''Task 1.''' Solve the following differential equations:
	# <math>x^2y^3+x(1+y^2)y'=0</math> (hint: try <math>\mu=x^\alpha y^\beta</math>).		# <math>x^2y^3+x(1+y^2)y'=0</math> (hint: try <math>\mu=x^\alpha y^\beta</math>).
	# <math>dx+(\frac{x}{y}-\sin y)dy=0</math>.		# <math>dx+(\frac{x}{y}-\sin y)dy=0</math>.
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	# <math>\frac{dy}{dx}=\frac{2y+\sqrt{x^2-y^2}}{2x}</math>.		# <math>\frac{dy}{dx}=\frac{2y+\sqrt{x^2-y^2}}{2x}</math>.

	'''~~Question~~ 2.''' Let <math>M</math> and <math>N</math> be differentiable functions of <math>x</math> and <math>y</math>.		'''Task 2.''' Let <math>M</math> and <math>N</math> be differentiable functions of <math>x</math> and <math>y</math>.
	# Show that if <math>\frac{N_x-M_y}{xM-yN}</math> depends only on <math>xy</math>, then the differential equation <math>Mdx+Ndy=0</math> has an integrating factor of the form <math>\mu(xy)</math>, where <math>\mu</math> is a function of a single variable.		# Show that if <math>\frac{N_x-M_y}{xM-yN}</math> depends only on <math>xy</math>, then the differential equation <math>Mdx+Ndy=0</math> has an integrating factor of the form <math>\mu(xy)</math>, where <math>\mu</math> is a function of a single variable.
	# Find a condition on <math>M</math> and <math>N</math> that would imply that the differential equation <math>Mdx+Ndy=0</math> would have an integrating factor of the form <math>\mu(x+y)</math>, where <math>\mu</math> is a function of a single variable.		# Find a condition on <math>M</math> and <math>N</math> that would imply that the differential equation <math>Mdx+Ndy=0</math> would have an integrating factor of the form <math>\mu(x+y)</math>, where <math>\mu</math> is a function of a single variable.

	'''~~Question~~ 3.''' The equation <math>y'+p(x)y=q(x)y^n</math> is called a "Bernoulli Equation".		'''Task 3.''' The equation <math>y'+p(x)y=q(x)y^n</math> is called a "Bernoulli Equation".
	# Explain why you already know how to solve the Bernoulli equation when <math>n=0</math> and when <math>n=1</math>.		# Explain why you already know how to solve the Bernoulli equation when <math>n=0</math> and when <math>n=1</math>.
	# Show that if <math>n\neq0,1</math>, then the substitution <math>v=y^{1-n}</math> reduces the Bernoulli equation to an equation you already know how to solve.		# Show that if <math>n\neq0,1</math>, then the substitution <math>v=y^{1-n}</math> reduces the Bernoulli equation to an equation you already know how to solve.
	# Solve the equation <math>x^2y'+2xy-y^3=0</math> (for <math>x>0</math>).		# Solve the equation <math>x^2y'+2xy-y^3=0</math> (for <math>x>0</math>).

	'''~~Question~~ 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.		'''Task 4.''' Find an example of a non-differentiable function which is nevertheless Lipschitz.