#$&*
course Mth 279
5/5
Query 17 Differential Equations*********************************************
Question: Solve the equation
25 y '' + 20 y ' + 4 y = 0, y(5) = 4 e^-2, y ' (5) = -3/5 e^-2
with y(5) = 4 e^-2 and y ' (5) = -3/5 e^-2.
Our characteristic equation is
25 r^2 + 20 r + 4 = 0
with repeated solution r = -2/5
yielding solution set
{e^(-2/5 t), t e^(-2/5 t) }
and general solution
y(t) = A e^(-2/5 t) + B t e^(-2/5 t)
satisfying
y(5) = 4 e^-2 and y ' (5) = -3/5 e^-2.
These conditions lead to the equations
A e^-2 + 5 B e^-2 = 4 e^-2
-2/5 A e^(-2) - 2/5 * 5 B e^-2 + B e^-2 = -3/5 e^-2,
or dividing both equations through by e^-2, and multiplying through the second by 5
A + 5 B = 4
-2 A - 5 B = -3.
The solution is A = 1, B = 3/5 ...
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
Working through this problem the above solution seems to hold true,
and would be a bit redundant to type again.
y(t) = e^(-2t/5) + (3t/5)e^(-2t/5)
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
1
.............................................
Given Solution:
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
------------------------------------------------
Self-critique rating:
*********************************************
Question: Solve the equation
3 y '' + 2 sqrt(3) y ' + y = 0, y(0) = 2 sqrt(3), y ' (0) = 3
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
Using the Quadratic Equation:
r_1,2 = (-2*sqrt(3)) +,- sqrt((2*sqrt(3))^2 -4(3)(1)))/2(3)
r1 = (-2*sqrt(3))/12
@&
-2 sqrt(3) / (2 * 3) = -sqrt(3) / 3.
*@
and r2 = r1
General solution:
y(t) = Ae^(-2*sqrt(3)/12 t) + Bte^(-2*sqrt(3)/12 t)
y'(t) = A(-2*sqrt(3)/12)e^(-2*sqrt(3)/12 t) + B(1 - (-2*sqrt(3)/12 t))e^(-2*sqrt(3)/12 t)
y(0) = Ae^0 +B(0)e^0 = 2*sqrt(3)
Ae^0 = 2*sqrt(3)
A = 2*sqrt(3)
y'(0) = A(-2*sqrt(3)/12)e^0 + B(1 - 0)e^0 = 3
y'(0) = A(-2*sqrt(3)/12) + B = 3
y'(0) = (2*sqrt(3))(-2*sqrt(3)/12) + B = 3
1 + B = 3
B = 2
y(t) = (2*sqrt(3))e^(-2*sqrt(3)/12 t) + 2te^(-2*sqrt(3)/12 t)
@&
Good, except for the early arithmetic error. Compare with the following solution:
The characteristic equation is
3 r^2 + 2 sqrt(3) r + 1 = 0
with repeated solution
r = (-2 sqrt(3) +- sqrt( 12 - 12 ) ) / 6 = -sqrt(3)/3.
This yields fundamental set
{e^(-sqrt(3) / 3 * t), t e^(-sqrt(3) / 3 * t) }
with general solution
y(t) = A e^(-sqrt(3) / 3 * t) + B t e^(-sqrt(3) / 3 * t).
y(0) = A = 2 sqrt(3)
y ' (0) = -sqrt(3) / 3 * A + B = 3.
Substituting A = 2 sqrt(3) into the second equation we get
-6 + B = 3
so that
B = 9.
Our solution is therefore
y(t) = 2 sqrt(3) e^(-sqrt(3) / 3 * t) + 9 t e^(-sqrt(3) / 3 * t).
*@
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
2
.............................................
Given Solution:
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
------------------------------------------------
Self-critique rating:
*********************************************
Question: Solve the equation
y '' - 2 cot(t) y ' + (1 + 2 cot^2 t) y = 0,
which has known solution y_1(t) = sin(t)
You will use reduction of order, find intervals of definition and interval(s) where the Wronskian is continuous and nonzero.
See your text for a more complete statement of this problem.
YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY
Your solution:
y2(t) = y1(t)u(t) = sin(t)u(t)
y '' - 2 cot(t) y ' + (1 + 2 cot^2 t) y = 0
(sin(t)u(t))'' - 2 cot(t) (sin(t)u(t))' + (1 + 2 cot^2 t)(sin(t)u(t)) = 0
y '' - 2 cot(t) y ' + (1 + 2 cot^2 t) y = sin(t)u(t)
u'' = 0, u(t) = a1*t + a2
y2(t) = sin(t)(a1*t + a2) = a1*tsin(t) + a2*sin(t)
let a2 = 0
y2(t) = a1*tsin(t)
y(t) = Asin(t) + B*t*sin(t)
The wronskian confirms that y1, y2 are a fundamental set.
W(t) = |f(t) g(t)| =f(t)g'(t) - g(t)f '(t)
|f '(t) g'(t)|
W(t) = |sin(t) tsin(t) |
|cos(t) sin(t) + tcos(t) |
= Sin^2(t) + tsin(t)cos(t) - tsin(t)cos(t) = sin^2(t)
confidence rating #$&*:
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
3
.............................................
Given Solution:
&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&
Self-critique (if necessary):
------------------------------------------------
Self-critique rating: