DIFFERENTIAL EQUATIONS EXAM
QUESTIONS WIT ALL CORRECT
ANSWERS
Two questions arise when approaching a DE - Answer-Does it have a solution and is it
unique?
In the logistic model for population growth dP/dt=P(8-2P), the carrying capacity of the
population is - Answer-k=a/b so in this case, 4
What is the difference between an IVP and a BVP? - Answer-an initial value problem
has all of the conditions specified at the same value of the independent variable in the
equation. ex, y(0)= #, y'(0)= # whereas a BVP may have y(0)= # y'(2)= #
When the Wronskian = 0, then is it linear dependent or independent? - Answer-Linear
Dependent
List the steps to these directions: y=c1x +c2xlnx is a solution of xy''-xy'=0 on (0,inf). Find
a member of the family of solutions that satisfies the initial conditions y(1)=5 and y'(1)=3
- Answer-Find y'
Plug in the initial condition to solve for the constants c1 and c2
List the steps to these directions: Determine whether the given set of functions is
linearly independent or linearly dependent - Answer-Use the Wronskian to determine
whether the functions or independent or dependent
List the steps to these directions: Verify that the given functions form a F.S.S of the
differential equation on the indicated interval. Form the general solution. - Answer-Set y
equal to each value given
Find the number of derivatives to match the order of the DE
Plug the derivatives into the given equation and verify that they each equal zero, which
makes them a F.S.S
Use the Wronskian to determine if they equal zero
List the steps to these directions: Verify that the given two- parameter family of functions
is the general solution of the non homogeneous differential equation on the indicated
interval - Answer-Let y1 equal the stuff behind c1 and y2 equal the stuff behind c2
Derive as many times as need to see if it satisfies the given equation
Take the Wronskian of y1 and y2, respectively in the columns
To be a F.S.S, the Wronskian cannot equal zero
, BVP - Answer-Linear differential equation of 2nd order or greater in which the
dependent variable y or its derivatives are specified at different points
Corollaries to the superposition principle - Answer-1) a constant multiple y=c1y1(x) of a
solution y1(x) of a homogeneous linear DE is also a solution
2) A homogeneous linear DE always possesses the trivial solution y=0
The two fundamental solutions that arise when considering an IVP - Answer-1) Does a
solution exist?
2) If a solution exists, is it unique?
True or False? L is the symbol for a linear operator - Answer-True
When discussing the logistics equation, we talked about the density dependent
hypothesis. This DE is given by dP/dt=Pf(P). What is the simpliest assumption we can
make about f(P)? - Answer-f(P) is linear
True or False? An initial Value Problem is guaranteed a solution that is unique and
exists - Answer-True
What are the two important assumptions when stating definitions and theorems about
linear equations on some interval I? - Answer-1) The coefficient functions
ai(x),i=0,1,2,...,n and g(x) are continuous
2) an(x) cannot equal zero for every x in the interval
True or False? Any linear differential equation can be expressed in terms of the D
notation - Answer-True
Superposition Principle- Homogeneous Equations states that... - Answer-The sum, or
superposition, of two or more solutions of a homogeneous linear DE is also a solution
How do you find the general solution to a Homogeneous DE? - Answer-Find the
Wronskian. If the Wronskian equals a number, then the given y values form a F.S.S
True or False? Every homogeneous linear differential equation always has a solution -
Answer-True
True or False? In the growth model dP/dt=50e^0.5t, the growth constant is 50 - Answer-
False
True or False? In the competition model dx/dt = ax-bxy, dy/dt=cy-dxy, where x(t) and
y(t) are the popultions of the competing species moose and deer, respectively, the
coefficient c represents the moose growth rate - Answer-False
True or False? The Wronskian of the function e^x and e^3x is 2e^2x - Answer-False
QUESTIONS WIT ALL CORRECT
ANSWERS
Two questions arise when approaching a DE - Answer-Does it have a solution and is it
unique?
In the logistic model for population growth dP/dt=P(8-2P), the carrying capacity of the
population is - Answer-k=a/b so in this case, 4
What is the difference between an IVP and a BVP? - Answer-an initial value problem
has all of the conditions specified at the same value of the independent variable in the
equation. ex, y(0)= #, y'(0)= # whereas a BVP may have y(0)= # y'(2)= #
When the Wronskian = 0, then is it linear dependent or independent? - Answer-Linear
Dependent
List the steps to these directions: y=c1x +c2xlnx is a solution of xy''-xy'=0 on (0,inf). Find
a member of the family of solutions that satisfies the initial conditions y(1)=5 and y'(1)=3
- Answer-Find y'
Plug in the initial condition to solve for the constants c1 and c2
List the steps to these directions: Determine whether the given set of functions is
linearly independent or linearly dependent - Answer-Use the Wronskian to determine
whether the functions or independent or dependent
List the steps to these directions: Verify that the given functions form a F.S.S of the
differential equation on the indicated interval. Form the general solution. - Answer-Set y
equal to each value given
Find the number of derivatives to match the order of the DE
Plug the derivatives into the given equation and verify that they each equal zero, which
makes them a F.S.S
Use the Wronskian to determine if they equal zero
List the steps to these directions: Verify that the given two- parameter family of functions
is the general solution of the non homogeneous differential equation on the indicated
interval - Answer-Let y1 equal the stuff behind c1 and y2 equal the stuff behind c2
Derive as many times as need to see if it satisfies the given equation
Take the Wronskian of y1 and y2, respectively in the columns
To be a F.S.S, the Wronskian cannot equal zero
, BVP - Answer-Linear differential equation of 2nd order or greater in which the
dependent variable y or its derivatives are specified at different points
Corollaries to the superposition principle - Answer-1) a constant multiple y=c1y1(x) of a
solution y1(x) of a homogeneous linear DE is also a solution
2) A homogeneous linear DE always possesses the trivial solution y=0
The two fundamental solutions that arise when considering an IVP - Answer-1) Does a
solution exist?
2) If a solution exists, is it unique?
True or False? L is the symbol for a linear operator - Answer-True
When discussing the logistics equation, we talked about the density dependent
hypothesis. This DE is given by dP/dt=Pf(P). What is the simpliest assumption we can
make about f(P)? - Answer-f(P) is linear
True or False? An initial Value Problem is guaranteed a solution that is unique and
exists - Answer-True
What are the two important assumptions when stating definitions and theorems about
linear equations on some interval I? - Answer-1) The coefficient functions
ai(x),i=0,1,2,...,n and g(x) are continuous
2) an(x) cannot equal zero for every x in the interval
True or False? Any linear differential equation can be expressed in terms of the D
notation - Answer-True
Superposition Principle- Homogeneous Equations states that... - Answer-The sum, or
superposition, of two or more solutions of a homogeneous linear DE is also a solution
How do you find the general solution to a Homogeneous DE? - Answer-Find the
Wronskian. If the Wronskian equals a number, then the given y values form a F.S.S
True or False? Every homogeneous linear differential equation always has a solution -
Answer-True
True or False? In the growth model dP/dt=50e^0.5t, the growth constant is 50 - Answer-
False
True or False? In the competition model dx/dt = ax-bxy, dy/dt=cy-dxy, where x(t) and
y(t) are the popultions of the competing species moose and deer, respectively, the
coefficient c represents the moose growth rate - Answer-False
True or False? The Wronskian of the function e^x and e^3x is 2e^2x - Answer-False
