Prof. S. Brick Diﬀerential Equations Exam 1 Math 238
Spring ’06 section 101
Print your name:
Show all of your work, and explain your reasoning.
1. Solve y0=xy2+x
2. Draw the phase diagram of the autonomous diﬀerential equation y0=y(5 −y)(y+ 2)2
and use it to sketch solution curves. (Here x≥0, but ymay be negative.) Identify and
classify the types of equilibrium solutions.
3. The vertical motion of an object near the surface of the Earth is subject to two forces:
a downward gravitational force FG=−mg (where mis the mass of the object, gis
the gravitational constant, and the negative sign represents the downward direction) and a
force FRdue to air resistance. Assuming that the force due to air resistance is proportional
to the velocity and using the fact that the sum of the forces is equal to the product of
mass and acceleration, set up a diﬀerential equation for the velocity. Mention the signs of
any other constants you use, explaining your reasoning.
4. The function y=Ce−x+x−1, where Cis a constant, is a solution to y0=x−y(you
need not check that). Find Csuch that the initial condition y(0) = 10 is satisﬁed.
5. Solve y0+2y
6. Solve cos(x) + ln(y)dx +√y+x
ydy = 0. You may give an implicit form for your
7. Using an appropriate substitution, transform xy0+ 7x3y= (x2+ 9)y1
3into either a ﬁrst
order linear equation or a separable equation. Explicitly mention the substitution and
whether the result is linear or separable. Do not solve.
8. Using an appropriate substitution, transform homogeneous equation xy2y0=x3+y3.
into either a ﬁrst order linear equation or a separable equation. Explicitly mention the
substitution and whether the result is linear or separable. Do not solve.