Exam 1

Question 1

derivative of tan(x)

Answer 1

= sec^2(x)

Question 2

derivative of sec(x)

Answer 2

= sec(x)tan(x)

Question 3

derivative of csc(x)

Answer 3

= -csc(x)cot(x)

Question 4

derivative of cot(x)

Answer 4

= -csc^2(x)

Question 5

integration of sec^2(x)

Answer 5

= tan(x)

Question 6

integration of csc^2(x)

Answer 6

= -cot(x)

Question 7

integration of sec(x)tan(x)

Answer 7

= sec(x)

Question 8

integration of csc(x)cot(x)

Answer 8

= -csc(x)

Question 9

integration of tan(x)

Answer 9

= ln(sec(x))

Question 10

integration of cot(x)

Answer 10

= -ln(csc(x)) or =ln(sin(x))

Question 11

Linear functions

Answer 11

functions of independent variable only. the dependent variable and all of its derivatives are of the first degree

no trig, no e^(independent variable), no ln(independent variable), no powers other than 1 on independent variable

Question 12

Theorem: Existence and Uniqueness of Solution (Given the IVP)

Answer 12

if f and partial y of f are defined for (x(not), y(not)) then the IVP has a unique solution

Question 13

How to solve linear DE

Answer 13

Linear Problem in standard form is
dy/dx + p(x) y = Q(x)

1) Find integrating factor mu(x) = e^integral(p(dx))
2) (d/dx)[mu(x)y] = mu(x)Q(x)
3) Integrate both sides with respect to x (DONT FORGOT CONSTANT OF INTEGRATION)
4) Rearrange to solve for desired variable

Question 14

Separable functions

Answer 14

A DE is separable if it can be written as
dy/dx = g(x)p(y)

Question 15

How to solve separable DE

Answer 15

standard form: dy/dx = g(x)p(y)

1. Check if zeros of p(y) are solutions of equation
2. divide both sides by p(y) and multiply both sides by dx to get form: (1/p(y))dy = g(x)dx
3. Integrate both sides to get an implicit solution

Question 16

Exact Equations

Answer 16

M(x,y)dx+N(x,y)dy = 0
(partial f or x)dx + (partial f of y)dy = 0

Question 17

Solving Exact equations

Answer 17

M(x,y)dx + N(x,y)dy = 0
1. Check for exactness partial M of y = partial N of x
2. If it is not exact, solve for integrating factor using
either mu(x) = e^integral ((M partial y - N partial x)/N) dx
or mu (y) = e^integral ((N partial x - M partial y)/M)dy
3. If/when exact f(x,y) = integral M(x,y)dx + g(y)
4. Take partial with respect to y
5. Set g’(y) = N and integrate
6. Solution f(x,y) = integral M(x,y)dx + g(y) = C

Question 18

Test for exactness

Answer 18

M partial y = n partial x

Question 19

Homogenous Equations

Answer 19

If the expression can have x/y or y/x or some factor of them or constants then it’s homogenous. Every x must have a y and every y must have a x

Question 20

Solving homogenous equations

Answer 20

1. Substitute y/x with v
   Now the de should be dy/dx = G(v)
2. we need to make dy/dx in terms of v and x
   v = y/x so y=vx and dy/dx = x (dv/dx) + v
3. sub this into separate variable and integrate
4. replace v with its respected terms of x and y

Question 21

Bernoulli Equations

Answer 21

When an equation looks linear but y is raised to a power
dy/dx + P(x) y = Q(x) y^n when n > 1

Question 22

Solving Bernoulli equations

Answer 22

1. note value of n
2. note value of 1-n
3. v = y^(1-n)
4. dv/dx = (1-n)y^n
5. use magic equaiton
   dV/dx + (1-n) P(x) v = (1-n)Q(x)