# Phase Diagram - Differential Equations - Exam, Exams for Differential Equations. Ankit Institute of Technology and Science

## Differential Equations

Description: Some keywords in Differential Equations are Convolution, Laplace Transform, Implicit Solution, Initial Condition, Integrating Factor, Autonomous Differential Equation, Appropriate Substitution. Some points of this exam paper are: Phase Diagram, Solve, Autonomous Differential Equation, Negative, Identify, Sketch Solution Curves, Equilibrium Solutions
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Prof. S. Brick Diﬀerential Equations Exam 1 Math 238
Spring ’06 section 101
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 x0, 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=Cex+x1, where Cis a constant, is a solution to y0=xy(you
need not check that). Find Csuch that the initial condition y(0) = 10 is satisﬁed.
5. Solve y0+2y
x=x.
6. Solve cos(x) + ln(y)dx +y+x
ydy = 0. You may give an implicit form for your
solution.
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.