1) Calculus

Question 1

What is Leibniz, Lagrange and Newton Notaition

Answer 1

Question 2

What is an anti-derivative

Answer 2

F′(x) = f(x)

Same as an integral

Question 3

What is an Explicit ODE

Answer 3

The Ode is written as

Otherwise it is implicit

Question 4

What is the order of an ODE

Answer 4

The order of the highest derivative appearing in the ODE

Question 5

What is a Linear ODE

Answer 5

An ODE which’s coefficiants of the derivatives of y only depend on x or a constant, otherwise the ODE is nonlinear

Question 6

What is an autonomous ODE

Answer 6

The ODE is autonomous if it does not depend on x. Otherwise it is non-autonomous.

Question 7

When is an ODE seperable

Answer 7

Question 8

What is Peano’s Existence Theorem

Answer 8

If f is continuous in [a, b]×[c, d]
Then there exists a local solution y : [x0 −δ, x0 +δ] → [c, d] for some δ > 0
We are not guaranteed existence of a global solution y : [a, b] → [c, d]

Question 9

What is Picard’s Uniqueness Theorem

Answer 9

Suppose that y : [a, b] → [c, d] solves the a initial value problem
If f and ∂f / ∂y are continuous in [a, b] × [c, d], then this solution is
unique.