DIFFERENTIAL EQUATIONS
COMPREHENSIVE EXAM SCRIPT VERIFIED
QUESTIONS AND SOLUTIONS GRADED
APLUS
●● What is the process for a separable equation?
Answer: Separate variables, integrate both sides, solve for y if possible,
then use the initial condition if given.
●● What does y' + P(x)y = Q(x) mean?
Answer: It is a first-order linear differential equation.
●● What formula do you use for first-order linear equations?
Answer: Use the integrating factor mu = e^(integral P(x) dx).
●● What are the steps for first-order linear equations?
Answer: Put in form y' + P(x)y = Q(x), find mu, multiply the whole
equation by mu, rewrite left side as (mu y)', integrate, then solve for y.
●● What is the biggest mistake with integrating factor problems?
Answer: Forgetting to put the equation in standard form y' + P(x)y =
Q(x) first.
, ●● What is a Bernoulli equation?
Answer: A first-order equation in the form y' + P(x)y = Q(x)y^n.
●● What substitution do you use for Bernoulli equations?
Answer: Use v = y^(1-n).
●● How do you recognize an autonomous equation?
Answer: It looks like dy/dt = f(y), meaning the right side only depends
on y.
●● How do you find equilibrium solutions?
Answer: Set the right side equal to 0 and solve.
●● How do you classify stability on a phase line?
Answer: Test signs between equilibrium points. Arrows pointing in
means stable. Arrows pointing out means unstable.
●● What does y' > 0 mean on a phase line?
Answer: The solution is increasing, so the arrow points right/up.
●● What does y' < 0 mean on a phase line?
Answer: The solution is decreasing, so the arrow points left/down.
COMPREHENSIVE EXAM SCRIPT VERIFIED
QUESTIONS AND SOLUTIONS GRADED
APLUS
●● What is the process for a separable equation?
Answer: Separate variables, integrate both sides, solve for y if possible,
then use the initial condition if given.
●● What does y' + P(x)y = Q(x) mean?
Answer: It is a first-order linear differential equation.
●● What formula do you use for first-order linear equations?
Answer: Use the integrating factor mu = e^(integral P(x) dx).
●● What are the steps for first-order linear equations?
Answer: Put in form y' + P(x)y = Q(x), find mu, multiply the whole
equation by mu, rewrite left side as (mu y)', integrate, then solve for y.
●● What is the biggest mistake with integrating factor problems?
Answer: Forgetting to put the equation in standard form y' + P(x)y =
Q(x) first.
, ●● What is a Bernoulli equation?
Answer: A first-order equation in the form y' + P(x)y = Q(x)y^n.
●● What substitution do you use for Bernoulli equations?
Answer: Use v = y^(1-n).
●● How do you recognize an autonomous equation?
Answer: It looks like dy/dt = f(y), meaning the right side only depends
on y.
●● How do you find equilibrium solutions?
Answer: Set the right side equal to 0 and solve.
●● How do you classify stability on a phase line?
Answer: Test signs between equilibrium points. Arrows pointing in
means stable. Arrows pointing out means unstable.
●● What does y' > 0 mean on a phase line?
Answer: The solution is increasing, so the arrow points right/up.
●● What does y' < 0 mean on a phase line?
Answer: The solution is decreasing, so the arrow points left/down.
