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course MTH 279
3.2Find at least an implicit solution to each equation. When possible, find an explicit solution and determine the t interval on which that solution exists:
1. dy/dt = 1 / y^4, y(1) = 4
y^4 dy = dt
Integrate
1/5 y^5 + C = t
1/5 (4)^5 + C = 1
C = .004883
1/5(y)^5 + .004883 = t
2. y ' + t y ^2 = 0, y(0) = 1.
dy/dt = -t * y^2
dy/y^2 = -t dt
Integrate
1/y = t^2 / 2 + C
1 = 0^2 / 2 + C
C = 1
1 = (y* t^2)/2 + y
3. y ' = 1 + y^2, y(pi/3) = sqrt(3).
dy/dt = 1 + y^2
dy/ (1+y^2) = dt
Integrate
ArcTan(y) + C = t
ArcTan(sqrt(3)) + C = pi/3
C is negligible at approximately - 2 * 10^-10
ArcTan(y) = t
4. y ' = e^(y - 2 t), y(0) = 1.
dy/dt = e^(y-2t)
ln(y)dy - y = -2t dt
Integrate
y(ln(y) -1) - y^2 / 2 = -t^2 + C
1(ln(1) -1) - 1^2 / 2 = 0^2 + C
-1 - ½ = C
C = -3/2
y(ln(y) -1) - y^2 / 2 = -t^2 -3/2
5. y ' + e^t sin^2(y) = 0, y(0) = pi/6.
dy/dt = - e^t sin^2(y)
1/ (sin^2 (y)) dy = e^t dt
Integrate
-cot(y) + C = e^t
-cot(pi/6) + C = e^(0)
C = 1.732 + 1
C = 2.732
-cot(y) + 2.732 = e^t
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Good, but I can't seem to locate these problems under Assignment 7.
They aren't in the QA, or the Query, or the assigned problems. Nor are they in the Class Notes.
Can you clarify?
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