Name
Differential Equations and Linear Algebra 2250
Midterm Exam 1 [Version 3]
Thursday, 1 October 2009
Instructions: This in-class exam is 50 minutes. No calculators, notes, tables or books. No
answer check is expected. Details count 3/4, answers count 1/4.
1. (Quadrature Equations)
3 + x2
(a) [25%] Solve y 0 =
.
1 + 2x + x2
(b) [25%] Solve y 0 = (sin x + sec x)(sec x tan x + cos x).
(c) [25%] Solve y 0 = sec (2x + 1) tan (2x + 1), y(0) = 2.
(d) [25%] Find the position x(t) from the velocity model
the position model
dx
= v(t), x(0) = 20.
dt
d −2t
(e v(t)) = 0, v(0) = 4 and
dt
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2250 Exam 1 F2009 [Ver 3]
Name.
2. (Classification of Equations)
The differential equation y 0 = f (x, y) is defined to be separable provided f (x, y) =
F (x)G(y) for some functions F and G.
(a) [40%] Check ( X ) the problems that can be put into separable form. No details expected.
y 0 + xy 3 = y(xy 2 + ex ) + x3 y
y 0 = (−x − 1)(y + 1) + xy + x + 2
y 0 = e2x (2ex−y e3y + 3e3x+2y )
y 0 + ey = ex+y
(b) [10%] Is y 0 + 2x y = yex separable? No details expected.
(c) [10%] Give an example of y 0 = f (x, y) which is linear and separable but not quadrature.
No details expected.
(d) [40%] Apply tests to show that y 0 = x3 + e2y is not separable and not linear. Supply
all details.
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2250 Exam 1 F2009 [Ver 3]
Name.
3. (Solve a Separable Equation)
Given (x + 1)(y + 3)y 0 = (x + 1)e−x+2 + 3x2 (y − 1)2 .
Find a non-equilibrium solution in implicit form.
To save time, do not solve for y explicitly and do not solve for equilibrium solutions.
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2250 Exam 1 F2009 [Ver 3]
Name.
4. (Linear Equations)
(a) [60%] Solve the linear model 5x0 (t) = −160 +
factor steps.
25
x(t), x(0) = 16. Show all integrating
t+2
dy
+ (3x2 + 1)y = 0.
dx
dy
(c) [20%] Solve 2 + 5y = 17 using the superposition principle y = yh + yp . Expected are
dx
answers for yh and yp .
(b) [20%] Solve the homogeneous equation
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2250 Exam 1 F2009 [Ver 3]
Name.
5. (Stability)
(a) [50%] Draw a phase line diagram for the differential equation
dx
= e2x ln(1 + 5x4 ) (3 − |x − 2|)3 (1 + x)(2 − x)(4 − x2 )3 .
dt
Expected in the phase line diagram are equilibrium points and signs of dx/dt.
(b) [50%] Assume an autonomous equation x0 (t) = f (x(t)). Draw a phase diagram with
at least 12 threaded curves, using the phase line diagram given below. Add these labels as
appropriate: funnel, spout, node [neither spout nor funnel], stable, unstable.
−
−
−
+
+
+
x
−12
−6
−3
0
3
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