Diff EQ Test 1

Question 1

1st order linear normal

Answer 1

1. divid so y’ has no coeff.
2. take integral of p(t) -> (the coeff. to y)
3. take integral of (e^(integral of p(t))*g(t)) ->
4. y(t) = e^-(integral p(t))*step 3

Question 2

1st order separable g(t)/h(t)

Answer 2

1. integral of h
2. set integral of h = integral of g
3. solve for y

Question 3

how to check if exact diff eq

Answer 3

if deriv with respect to y of the first part = deriv with respect to x of the second part

Question 4

solving exact diff eq

Answer 4

1. take integral of first part of eq.
2. take deriv of step 1 with respect to y
3. set second part of eq. = step 2
4. solve for h’
5. take integral of h’
6. plug h’ into step 2
7. solve for y

Question 5

2nd order homo easy diff eq

Answer 5

1. rewrite with r
2. find discriminant
3. find roots
4. gen sol: y = C1e^(r1t)+C2e^(r2t)

Question 6

2nd order homo diff eq with imaginary numbers

Answer 6

same steps as regular second order, but gen sol = C1e^(pt)cos(qt)+C2e^(pt)sin(qt) where if root = 1+2i, p=1 q=2

Question 7

2nd order homo diff eq with same roots

Answer 7

same steps, but gen sol = C1e^(rt)+C2te^(rt)

Question 8

variation of parameters

Answer 8

y=v1y1+v2y2+c1y1+c2y2

where v1 = - integral[(gy2)/w] and v2 = integral[(gy1)/w]

Question 9

laplace of e^at

Answer 9

1/(s-a)

Question 10

laplace of sin(at)

Answer 10

a/(s^2+a^2)

Question 11

laplace of cos(at)

Answer 11

s/(s^2+a^2)

Question 12

laplace of t^n

Answer 12

n!/t^(n+1)

Question 13

laplace of 1

Answer 13

1/s

Question 14

reduction of order

Answer 14

find y2=vy1 where v = integral(e^-integral(p(t))/y1^2)

Question 15

laplace of e^at*cos(bt)

Answer 15

(s-a)/[(s-a)^2+b^2]

Question 16

laplace of e^at*sin(bt)

Answer 16

b/[(s-a)^2+b^2]