Download Problem 1. Solve the initial-value problem dy dt = ty2 + 2y2 + t + 2 , y

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Transcript
Math 2326
First Midterm
Khamsi
Problem 1. Solve the initial-value problem
dy
= ty 2 + 2y 2 + t + 2 , y(0) = 1.
dt
Problem 2. Find the solution to
dy
= −5y + sin(t) , y(0) = 1.
dt
Problem 3. A cup of hot chocolate is initially at 110o F
and is left in a room with ambient temperature of 70o F .
Suppose that after one minute, the chocolate cooled
10o F . Write down the differential equation satisfied by
the temperature of the chocolate. Then solve it. How
long does it take the hot chocolate to cool to 75o F ?
Problem 4. Use Euler’s method with the given step size
∆t to approximate the solution to the given initial-value
problem over the time interval specified.
dy
= e− y/2 , y(0) = 2, 0 ≤ t ≤ 3, ∆t = 0.5
dt
Find the exact solution and determine the error at t = 3.
Draw the graph of the approximated solution as well as
the exact one.
1
Problem 5. For the one-parameter family
dy
= cos(y) + α
dt
identify the bifurcation values of α. Draw the phase
1 1
lines for α = −2, −1, − , , 1, 2.
2 2
Classify the constant solutions (if any) as sink, source,
1 1
or nodes for α = −2, −1, − , , 1, 2.
2 2
1 1
Draw some solutions for α = −2, −1, − , , 1, 2.
2 2
2