query 07

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course Mth 279

3/26/14

Query 07 Differential Equations*********************************************

Question: 3.4.2. Solve the equation y ' = 2 t y ( 1 - y), with y(0) = -1, as a Bernoulli equation.

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Your solution:

->Bernoulli equation form: y' + p(t)y = q(t)y^n

y ' = 2 t y ( 1 - y)

y ' = 2 t y - 2ty^2

y ' - 2 t y = - 2ty^2

This equation is now in the correct form.

p(t) = -2t

q(t) = -2t

n = 2

Using v = y^m, which means y = v^(1/m), and m = (1 - n) from the Bernoulli equation.

The equation becomes:

dv/dt + (1-n)p(t)v = (1 - n)q(t)

dv/dt + (1-2)(-2t)v = (1 - 2)(-2t)

dv/dt + (-1)(-2t)v = (- 1)(-2t)

dv/dt + (2t)v = (2t)

This Equation is now First Order linear, and can be more easily solved

dv/dt + (2t)v = (2t)

dv/dt = (2t) - (2t)v

dv/dt = (2t) (1 - v)

dv/(1 - v) = 2t dt

integrate:

ln(1 - v) = t^2 + c

1 - v = Ce^(t^2)

not sure if I took the correct derivative

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Good so far.

y = v^m.

You need to solve the equation for y.

You also need to apply the initial conditions.

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confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):OK

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Self-critique rating:

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Question: 3.4.6. Solve the equation y ' - y = t y^(1/3), y(0) = -9 as a Bernoulli equation.

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Your solution:

->Bernoulli equation form: y' + p(t)y = q(t)y^n

y ' - y = t y^(1/3)

Since the Equation is already in the form:

p(t) = -1

q(t) = t

n = 1/3

Using v = y^m, —> y = v^(1/m), and m = (1 - n) from the Bernoulli equation.

The equation becomes:

dv/dt + (1-n)p(t)v = (1 - n)q(t)

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The right-hand side of the equation is just q(t).

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dv/dt + (1-(1/3))(-1)v = (1 - (1/3))(t)

dv/dt + (2/3)(-1)v = (2/3)(t)

dv/dt + (- 2/3)v = (2/3)(t)

use u(t) = e^(int(p(t))dt) for the form y' + p(t)y = g(t)

For this equation:

p(t) = (-2/3)

int(p(t)) dt = (-2/3)t = -2t/3

u(t) = e^(-2t/3)

v' + (- 2/3)v = (2/3)(t)

v' e^(-2t/3) + (- 2/3)ve^(-2t/3) = (2/3)(t)e^(-2t/3)

v' e^(-2t/3) - (2/3)ve^(-2t/3) = (2/3)(t)e^(-2t/3)

(ve^(-2t/3))' = (2/3)te^(-2t/3)

Integration of both sides yeilds:

ve^(-2t/3) = Int ((2/3)te^(-2t/3)) dt

Integration of the Right hand side

udv = uv - Int(vdu)

u = t

dv = e^(-2t/3) dt

du = dt

v = Int(dv) = (-3/2)e^(-2t/3)

The Equation becomes:

ve^(-2t/3) = (2/3)((-3/2)te^(-2t/3) + (3/2) Int(e^(-2t/3)) dt)

ve^(-2t/3) = -te^(-2t/3) + (-3/2)e^(-2t/3)

ve^(-2t/3) = -te^(-2t/3) - (3/2)e^(-2t/3)

Dividing Out u(t) = e^(-2t/3), the Equation becomes:

v = -t - (3/2) + c

—> y = v^(1/m) = v^(1/(1 - n))

y = v^(1/(1 - 1/3)) = v^(3/2)

y = (-t - (3/2) + c)^(3/2)

y(0) = (-0 - (3/2) + c)^(3/2)

(- (3/2) + c)^(3/2) = -9

(- (3/2) + c)^(3) = 81

(- (3/2) + c) = cube root (81)

C = cube root (81) + (3/2) = 5.83 approx.

Equation becomes:

y = (-t - (3/2) + cube root (81) + (3/2))^(3/2)

y = (-t + cube root (81))^(3/2)

y = (cube root (81) - t)^(3/2)

confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):OK

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Self-critique rating:

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Question: 3.4.8. Solve the equation y ' = - (y + 1) + t ( y + 1)^(-2) as a Bernoulli equation.

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Your solution:

Is it possible to use substitution u=y+1?

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That is perfectly valid, and in this case it conveniently works out that du = dy.

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confidence rating #$&*:

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Given Solution:

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Self-critique (if necessary):Not sure on this one

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Self-critique rating:"

Self-critique (if necessary):

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Self-critique (if necessary):

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&#Good responses. See my notes and let me know if you have questions. &#