Mat 540 Final
MA1506 TUTORIAL 1
1.
Solve the following differential equations:
(a) x(x + 1)y = 1
(b) (sec(x))y = cos(5x)
(c) y = e(x−3y)
(d) (1 + y)y + (1 − 2x)y 2 = 0
Use www.graphmatica.com to sketch the functions you found as solutions of [a]-[d], if y =
1/2 at x = 1. Graphmatica can also directly sketch the graph if you insert the equation itself; for example, in [a] you just enter x(x + 1)dy = 1 {1, 1/2}. [curly brackets for the initial conditions, with x followed by y.] Here dy represents y . Use this to check that your answers were correct.
2. Experiments show that the rate of change of the temperature of a small iron ball is proportional to the difference between its temperature T (t) and that of its environment, Tenv (which is constant). …show more content…
Show that T = Tenv is a solution.
Does this make sense? The ball is heated to 300◦ F and then left to cool in a room at 75◦ F . Its temperature falls to 200◦ F in half an hour. Show that its temperature will be 81.6◦ F after 3 hours of cooling.
3. In very dry regions, the phenomenon called Virga is very important because it can endanger aeroplanes. [See http://en.wikipedia.org/wiki/Virga ]. Virga is rain in air that is so dry that the raindrops evaporate before they can reach the ground. Suppose that the volume of a raindrop is proportional to the 3/2 power of its surface area. [Why is this reasonable? Note: raindrops are not spherical, but let’s assume that they always have the same shape, no matter what their size may be.] Suppose that the rate of reduction of the volume of a raindrop is proportional to its …show more content…
What are the units of a(t)? The [first]
Friedmann Equation [http://en.wikipedia.org/wiki/Friedmann equations] relates this function to the energy density of the Universe and to its spatial curvature. In a particular cosmological model,
2
the Friedmann equation takes the form L2 a2 = a2 − a2 + 1, where L is a positive constant,
˙
the dot denotes time differentiation, and the initial condition is a(0) = 1. What are the units of
L? Show, without solving this equation, that the universe described by this model is never smaller than a certain minimum size. [Hint: graph the right side as a function of a, and note that the left side cannot be negative.] Now solve the equation and describe the history of this universe.
If you try to solve this equation using Graphmatica [www.graphmatica.com], you will find something very strange. [You have to type dy = sqrt(y 2 − (2/y 2 ) + 1){0, 1}.]
Is the solution given by Graphmatica correct? Try typing dy = sqrt(y 2 −(2/y 2 )+1){0, 1.00000000001} into graphmatica. You should see something much more like your solution.
If you prefer and already know how to use MATLAB, try using it too.
6.
Solve the following equations:
(a) y
1.
Solve the following differential equations:
(a) x(x + 1)y = 1
(b) (sec(x))y = cos(5x)
(c) y = e(x−3y)
(d) (1 + y)y + (1 − 2x)y 2 = 0
Use www.graphmatica.com to sketch the functions you found as solutions of [a]-[d], if y =
1/2 at x = 1. Graphmatica can also directly sketch the graph if you insert the equation itself; for example, in [a] you just enter x(x + 1)dy = 1 {1, 1/2}. [curly brackets for the initial conditions, with x followed by y.] Here dy represents y . Use this to check that your answers were correct.
2. Experiments show that the rate of change of the temperature of a small iron ball is proportional to the difference between its temperature T (t) and that of its environment, Tenv (which is constant). …show more content…
Show that T = Tenv is a solution.
Does this make sense? The ball is heated to 300◦ F and then left to cool in a room at 75◦ F . Its temperature falls to 200◦ F in half an hour. Show that its temperature will be 81.6◦ F after 3 hours of cooling.
3. In very dry regions, the phenomenon called Virga is very important because it can endanger aeroplanes. [See http://en.wikipedia.org/wiki/Virga ]. Virga is rain in air that is so dry that the raindrops evaporate before they can reach the ground. Suppose that the volume of a raindrop is proportional to the 3/2 power of its surface area. [Why is this reasonable? Note: raindrops are not spherical, but let’s assume that they always have the same shape, no matter what their size may be.] Suppose that the rate of reduction of the volume of a raindrop is proportional to its …show more content…
What are the units of a(t)? The [first]
Friedmann Equation [http://en.wikipedia.org/wiki/Friedmann equations] relates this function to the energy density of the Universe and to its spatial curvature. In a particular cosmological model,
2
the Friedmann equation takes the form L2 a2 = a2 − a2 + 1, where L is a positive constant,
˙
the dot denotes time differentiation, and the initial condition is a(0) = 1. What are the units of
L? Show, without solving this equation, that the universe described by this model is never smaller than a certain minimum size. [Hint: graph the right side as a function of a, and note that the left side cannot be negative.] Now solve the equation and describe the history of this universe.
If you try to solve this equation using Graphmatica [www.graphmatica.com], you will find something very strange. [You have to type dy = sqrt(y 2 − (2/y 2 ) + 1){0, 1}.]
Is the solution given by Graphmatica correct? Try typing dy = sqrt(y 2 −(2/y 2 )+1){0, 1.00000000001} into graphmatica. You should see something much more like your solution.
If you prefer and already know how to use MATLAB, try using it too.
6.
Solve the following equations:
(a) y