DIFFERENTIAL EQUATIONS EXAM
QUESTIONS WITH ALL CORRECT
ANSWERS
What is a phase line? - Answer-A simplified slope field for autonomous equations.
In an autonomous equation, what does the slope look like? Why? - Answer-The slope
field is constant along horizontal lines, because it doesn't depend on t.
Do solutions cross? - Answer-No!
What do logistic equations model? - Answer-They model population growth with a pop.
cap.
What is the basic logistic equation model? Exponential and with cap? Simplified? -
Answer-x' = ax
x' = a(x(1 - x/n)
x'= x(1-x)
How do we model a logistic equation with a parameter (such as rabbit pop growth, while
the farmer kills a rabbit a day?) - Answer-x' = x(1-x) - h
What is a set of phase lines together called? - Answer-A phase portrait or a bifurcation
diagram
How to solve axˆ2 + bx + c = 0?? - Answer-x equals -b plus or minus square root of (bˆ2
- 4ac) all over 2a
What do we mean by bifurcation? What creates it? - Answer-The qualitative change in
the solution when the parameter changes in certain differential equations. It is created
by a new equilibrium appearing or an old one disappearing.
How do we solve equations of the form x'= polynomial(x)? LIke x'= x(1-x) - Answer-dx/dt
= x(1-x) -- ∫dx/x(1-x) = ∫dt -- ∫dx/x-xˆ2 = t + c -- [to find ∫dx/p(x), use partial fractions] --
∫dx/x(1-x) = A/x + B/(1-x) = A(1-x) + B
Log(uv) = - Answer-Log(u) + Log(v)
Log(u/v) = - Answer-Log(u) - Log(v)
Log (u)ˆn = - Answer-nlog(u)
QUESTIONS WITH ALL CORRECT
ANSWERS
What is a phase line? - Answer-A simplified slope field for autonomous equations.
In an autonomous equation, what does the slope look like? Why? - Answer-The slope
field is constant along horizontal lines, because it doesn't depend on t.
Do solutions cross? - Answer-No!
What do logistic equations model? - Answer-They model population growth with a pop.
cap.
What is the basic logistic equation model? Exponential and with cap? Simplified? -
Answer-x' = ax
x' = a(x(1 - x/n)
x'= x(1-x)
How do we model a logistic equation with a parameter (such as rabbit pop growth, while
the farmer kills a rabbit a day?) - Answer-x' = x(1-x) - h
What is a set of phase lines together called? - Answer-A phase portrait or a bifurcation
diagram
How to solve axˆ2 + bx + c = 0?? - Answer-x equals -b plus or minus square root of (bˆ2
- 4ac) all over 2a
What do we mean by bifurcation? What creates it? - Answer-The qualitative change in
the solution when the parameter changes in certain differential equations. It is created
by a new equilibrium appearing or an old one disappearing.
How do we solve equations of the form x'= polynomial(x)? LIke x'= x(1-x) - Answer-dx/dt
= x(1-x) -- ∫dx/x(1-x) = ∫dt -- ∫dx/x-xˆ2 = t + c -- [to find ∫dx/p(x), use partial fractions] --
∫dx/x(1-x) = A/x + B/(1-x) = A(1-x) + B
Log(uv) = - Answer-Log(u) + Log(v)
Log(u/v) = - Answer-Log(u) - Log(v)
Log (u)ˆn = - Answer-nlog(u)
