Math 320 True False
Method of undetermined coefficients particular solution for cos(x)
Acos(x)+Bsin(x)
For a m x n matrix A, a vector in the nullspace is in R^m
False
The solution of the ODE (2+x^2)y"+(1-x)y=1 that satisfies the initial condition y(0)=0, y'(0)=0 is defined on the entire interval
(-infinity,infinity)
The solution of the ODE (2-x^2)y"+(1-x)y=1 that satisfies the initial condition y(0)=0, y'(0)=0 is defined on the entire interval
(-root2, root2)
Method of undetermined coefficients particular solution for 1+e^-x
Ae^-x+B
Method of undetermined coefficients particular solution form for x^2
Ax^2+Bx+C
How do you check if y1, y2, and y3 are linearly independent? ( Using stuff learned after 2nd exam)
Check that their Wronskian is nonzero
Every solution to a differential equation has at most a single vertical asymptote
False
For a m x n matrix A, a vector in the column space is in R^n
False
If AB=D and B is invertible, then A=B^-1D
False
If Ax=b and Ay=b, then x=y
False
If dy/dx = f(y) has a critical point y=c, then every solution satisfies y(x)->c as x->infinity
False
If dy/dx = f(y) has an unstable critical point, it must also have a stable critical point
False
If dy/dx=f(y) has a stable critical point, it must also have an unstable critical point
False
If dy/dx=f(y) has an unstable critical point y=c, then there is no solution such that y(x)->c as x->infinity
False
If dy/dx=f(y) has only one critical point y=c, then every solution satisfies y(x)->c as x->infinity
False
The columns of A always form a basis for the Column space of A
False
The differential equation y"-xy=0 has two linearly independent solutions for xE(-infinity,infinity) that satisfies the initial condition y(0)=0, y'(0)=1
False
The function y(x)=0 is one of many solutions of the differential equation y"+y'+y=0 that satisfies y(0)=0, y'(0)=0
False
The function y(x)=0 is the only solution of the differential equation y"+y'+y=0
False
The functions ln(x),ln(x^2),ln(x^3) are linearly independent on the interval (0,infinity)
False
The functions of x^3 and (abs.value)xx^2 are linearly independent on the interval (-infinity,0)
False
The functions x^3 and (abs. value)xx^2 are linearly independent on the interval (0,infinity)
False
The system Ax=b always has the solution x=0
False
The system Ax=b could have the solution x=0 when b!=0
False
The system Ax=b has a unique solution if b is in the column space of A
False
dy/dx=f(y) is guaranteed to have a critical point or equilibrium solution
False
yy'=x-1 is guaranteed to have only one solution passing through y(1)=0
False
For a m x n matrix A, a vector in the column space is in R^m
True
For a m x n matrix A, a vector in the nullspace is in R^n
True
The differential equation y"-xy=0 has two linearly independent solutions for xE(-infinity,infinity)
True
The function y(x)=0 is a solution of the differential equation y"+y'+y=0
True
The function y(x)=0 is the only solution of the differential equation y"+y'+y=0 that satisfies y(0)=0, y'(0)=0
True
The functions x^3 and (abs. value)xx^2 are linearly independent one the interval (-infinity,infinity)
True
The system Ax=0 always has the solution x=0.
True
The system Ax=b could have the solution x=0
True
dy/dx = f(y) may have infinitely many equilibrium solutions
True
y'=ln(1+y^2) is guaranteed to have only one solution satisfying y(0)=0
True
y'=ln(yx) is guaranteed to have only one solution satisfying y(1)=1
True
y'=x(y-1)^1/3 is guaranteed to have a solution satisfying y(0)=1
True
y'=xlny is guaranteed to have only one solution passing through y(0)=1
True
yy'=e^xsinx is guaranteed to have only one solution y(2pi)=1
True
yy'=x-1 is guaranteed to have only one solution passing through y(0)=1
True
When is a differential equation in equilibrium?
When dy/dx=0
Suppose A and B are invertible matrices, then the inverse matrix of AB is
b^-1a^-1
If y'=y(1-y) with y(0)=1, then y(1)
is equal to 1
If y'=y+cosx with y(0)=1, then y(1)
is greater than 1
If y'=y+sinx with y(0)=1, then y(1)...
is greater than 1
If y'=y with y(0)=-1, then y(1)
is less than -1
If y'=(y-2)y with y(0) = 1, then y(1)...
is less than 1
If detA!=0, the system Ax=b always has
one solution
if detA=0, then Ax=b always has
possibly no solution
Axis to graph to check stability of solution in equilibrium problem.
x' vs. x or y' vs. y depending on the variable used
Euler's Method
y1=y0+hf(x0,y0) y2=y1+hf(x1,y1) h=step size
