#$&*
Mth 279
Your 'question form' report has been received. Scroll down through the document to see any comments I might have inserted, and my final comment at the end.
** Question Form_labelMessages **
Homework #9 Problem 2 question
** **
Problem 2 y' - y = ty^(1/3) use bernoulli equation
In my submitted homework for Assignment 9 you had the following note:
To solve y ' + p(t) y = q(t) y^n:
1. Let v = y^m, where m = 1 - n.
2. The resulting equation will be v ' + m p(t) v = q(t).
3. Solve for v.
4. Plug in y^m for v.
In this case, n = 1/3 so m = 1 - n = 1 - 1/3 = 2/3.
v = y^m = y^(2/3)
The resulting equation is
v ' - 2/3 v = t
** **
What is confusing me is in your step 2 you have the resulting equation is v' + mp(t)v = q(t), where m = (1-n)?
In your lectures, you have the equation stated as v' + (1-n)p(t)v = (1-n)q(t). Why in you notes now is there no (1-n) or m in front of the q(t)? Because have that m and not having the m value in front of the q(t) changes the problem completely.
** **
If you use v' + (1-n)p(t)v = (1-n)q(t) instead the resulting equation would be v' - 2/3v = 2/3t instead of the equation you replied with v' - 2/3v = t
@&
The second step in my note should have read
"2. The resulting equation will be v ' + m p(t) v = m q(t)."
More detail below, based on the transformation v = y^m = y^(1-n):
*@
@&
Putting the equation in the form
y ' + p(t) y = q(t) y^n
we obtain
y ' - y / 2 = 1/2 t y^(1/3).
Letting v = y^m, with m = 1 - n = 1 - 1/3 = 2/3, we have
v = y^(2/3)
so that
v ' = 2/3 y^(-1/3) y ',
giving us
y ' = 3/2 y^(1/3) v ';
since v = y^(2/3), y(^1/3) = v^(1/2) so
y ' = 3/2 v^(1/2) v '.
Also y = v^(3/2) and y^(1/3) = v^(1/2).
Our equation therefore becomes
3/2 v^(1/2) v ' - 1/2 v^(3/2) = 1/2 t v^(1/2).
Dividing through by 3/2 v^(1/2) we get
v ' - 1/3 v = 1/3 t.
This linear nonhomogeneous equation is easily solved, using the usual integrating factor.
We then convert the solution for v into a solution for y.
*@
@&
Reconciling this with the symbols m and n and the p(t) and q(t) functions:
n = 1/3
m = 1 - n = 2/3
p(t) = -1/2
q(t) = 1/2 t
v ' + m p(t) v = q(t)
is therefore
v ' + 2/3 * (-1/2) v = 2/3 * 1/2 t
which simplifies to our equation
v ' - 1/3 v = 1/3 t.
*@
@&
This also clearly reconciles also with the form
v ' + (1 - n) p(t) v = (1 - n) q(t)
given in the videos.
*@
@&
I apologize for the omission in Step 2 of my note.
*@
@&
There is also a related error a few lines before Step 2, in the working out of the symbolic equation. Those lines should clearly read
"Multiplying both sides by m v^( (m - 1) / m) we have
v ' + m p(t) v = m q(t) v^(( n + m - 1) / m) ."
My carelessness in neglecting to multiply the right-hand side by m led to the error in Step 2.
*@