1.1, 1.2

Question 1

Integration by Parts Derivation

Answer 1

deriv ( xy ) = x’y + xy’
Integrate both sides
xy (from a to b) = int(x’y + xy’)
int(xy’) = x*y (from a to be) - int(x’y)
Inputs on the left side, outputs on the right

Question 2

Order of an Ordinary Differential Equation

Answer 2

largest n s.t. x^(n) appears in the equation

Question 3

an nth order ODE is linear if it can be put in the form:

Answer 3

a.n(t) * x^n + a.n-1(t) * x^n-1 + … + a.0(t) * x = r(t)
AKA on the left are x, x’, x’’ are all mult. by functions of t. On the right is a function of t.

Question 4

x’’ + tx^2 = e^(-t^2) is linear or nonlinear?

Answer 4

nonlinear b/c of the x^2

Question 5

Linear or not?
(1+sin(t))x’’ + x’ * e^t = t-x

Answer 5

Linear, because you can move the x to the other side and it fits the form.

Question 6

How do you verify a solution to an Ordinary Differential Equation

Answer 6

plug in the derivatives to the equation and see if it is still true

Question 7

What is an IVP problem (initial value problem)?

Answer 7

A differential equation that also has an initial condition, allowing you to solve for the value of C

Question 8

Fundamental Theorem of Existence and Uniqueness: x’ = f(x,t) , x(t.0) = x.0. Suppose f continuous at (t.0,x.0) and df/dx continuous at (t.0,x.0). Then:

Answer 8

there exists and interval a<t<b containing t.0 such that on this interval x has a unique solution. However, this does not tell you anything about the size of the interval

Question 9

A first order ODE is in the form:

Answer 9

x’ + p(t) * x = q(t)

Question 10

A linear equation is called homogeneous if:

Answer 10

if RHS is 0. AKA, r(t) = 0, q(t) = 0

Question 11

What is the solution to first order homogeneous equations (i.e. x’ + p(t)x = 0)?

Answer 11

Ce^(-P(t)), where P(t) is any antiderivative of p(t). This can be derived using integrating factor

Question 12

How do you solve first order inhomogeneous solutions (i.e. x’ + p(t) * x = q(t) )?

Answer 12

Integrating factor method!
1. u(t) = int(p(t))
2. Multiply equation by u(t)
3. Integrate to get solution, where LHS will collapse into a product rule.

Question 13

What is a particular homogeneous solution vs. a homogeneous solution (i.e. c(t + 1)^2 − (t + 1)^2 )?

Answer 13

Particular homogeneous solution does not include C constant (i.e. second half), and is unique up to adding copies of the homogeneous solution to it. AKA, this is a family of solutions and when c=0 that is the particular solution.

Question 14

When can you use Method of Undetermined Coefficients?

Answer 14

When equation is a constant coefficient equation ODE (i.e. x’ + px = q(t) ) NOT x’ + p(t)x = q(t)
AND
q(t)’s derivatives span finitely many functions

Question 15

How do you use Method of Undetermined Coefficients?

Answer 15

1. Create an Ansatz (guess) that is a constant times each of the spanned functions (i.e. x(t) = At^2 + Bt + C).
2. Plug in the guess into the equation and find a system of equations that matches the RHS.
3. Solve the system of equations to get the particular solution
4. Add the general solution: Ce^-pt

Question 16

An equation x’ = f(t,x) is called separable if:

Answer 16

if f(t,x) can be decomposed f(t,x) = g(t)h(x)

Question 17

How do you solve separable equations (i.e. x’ = f(t,x) = g(t)h(t) )?

Answer 17

1. x’ / h(x) = g(t)
2. If O(x) is an antiderivative of 1/h(x), then this is d/dt( O(x) ) = g(t)
3. O( x(t) ) = G(t) + c
4. Solve for x(t) if you can

Question 18

Span of a family of vectors or functions is:

Answer 18

All vectors or functions that can be made with linear combinations of original vectors