Math 320 True False

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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


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