110209 differential equations

`q006.  Show that the equation

( sqrt(y) - e^t cos(y) + t ) dt + (t / (2 sqrt(y) )+ e^t sin(y)) dy = 0

is of the form

N dt + M dy = 0, with N_y = M_t.

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Integrate N with respect to t, recalling that since y is treated as a constant, the integration constant can be any function f(y).

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Integrate M with respect to y, recalling that since t is treated as a constant, the integration constant can be any function g(t).

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Show that these integrals can be made equal by making a good choice of f(y) and g(t).

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Let F(t, y) be the common integral of the two functions.

Show that our equation is of the form dF = 0.

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Argue that the solution is therefore of the form F(t, y) = c, for a constant c.

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`q001.  Solve the equation

y ' = sin(t) / y, y(0) = 1.

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Over what region(s) of the plane does the existence theorem apply?

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What is the domain of your solution?

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`q002.  In general air resistance has components which depend on v and to v^2, where v is the velocity of an object.  If v ' =  -5 t + 4 t v + t v^2, what is the solution of the equation for which v(0) = 10? 

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Over what t interval can we expect this solution to exist?

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`q003.  Show that the equation

2 y tan(t) dy + (y^2 tan(t) sec (t) + 1 / (2 sqrt(t) ) dt = 0

is of the form M dy + N dt = 0, with M_t = N_y.

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Integrate to find the function F(t, y) for which F_t = N, and F_y = M.

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`q004.  Consider the equation y ' + p(t) y = q(t) y^3.

Let v(t) = y(t)^m, where the value of m has yet to be determined.

What is dv/dt?

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What therefore is dy/dt in terms of v and dv/dt?

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What is y in terms of v?

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Rewrite the original equation in terms of p(t), q(t), v and dv/dt.

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Choose m so that the v in the right-hand side has power 0.

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Now the equation is linear nonhomogeneous order 1. 

`q What integrating factor allows us to solve the equation?

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NOTE: This is an example of a Bernoulli Equation, which is of the form

y ' + p(t) y = q(t) y^n.

This equation differs from a first-order linear homogeneous equation by the factor y^n on the right-hand side.

The general method is to change the variable to v = y^m, then choose the value of m that makes the equation linear. 

`q005.  A population function P(t) satisfies dP/dt = k P, as long as space and resources are unlimited.  That is, the rate of growth is proportional to the population.

If, however, the population grows too much, it approaches the carrying capacity L of its environment, and its rate of growth becomes also proportional to (L - P).  Thus the rate of population growth is jointly proportional to P and (L - P).

This gives us the equation

dP/dt = k P ( L - P ).

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This equation is separable.  Find its general solution.  You will need to use partial fractions.

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If L = 1000, P(0) = 100 and P(1) = 200, what is the value of k?

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