Week 19

Question 1

Differential operator?

Answer 1

D(y)= dy/dx
D^2(y)=d^2y/dx^2

Question 2

Homogenous Differenctial eq if distinct real roots?

Answer 2

yx = A(e)^m1x +B(e)^m2x

Question 3

Homogenous Differential eq if repeated real roots?

Answer 3

yx = A+Bx+Cx)(e)^mx

Question 4

Homogenous Differential eq if complex conjugate pair?

Answer 4

yx= e^ax(C cos (bx) + D sin (bx))

where root is m=a+- ib

Question 5

Mimic cos(x)?

Answer 5

(PI)x = a cos x + b sin x

Question 6

Mimic exponential?

Answer 6

if (Q)x belongs to CS then multiply by axe^mx

but if it does not e.g. power is diff then we just multiply by x

Question 7

Seperable ODE look and how to solve?

Answer 7

dy/dx = F(X)G(y(x))

just move y term to LHS and dx to RHS and integrate

EXPLICIT FORM is just rearranging this term for y=…

Question 8

Exact ODE look and how to know if exact?

Answer 8

M(x,y) + N(x,y)dy/dx = 0

if My = Nx (partial derivatives are equal)

Question 9

How to solve exact ODE?

Answer 9

Integrate Fx =M(x,y) with respect to x and Fy = N(x,y) with respect to y
we get an eq for F from both
and compare to work out g(y) from 1 and h(x) from 2

If both g(y) = h(x) then we have constant K but final form is without and is just equal to c
then put in form Fx = c

Question 10

Linear ODE what they look like?

Answer 10

dy/dx + P(x)y = Q(x)

Question 11

How to solve linear ODE?

Answer 11

I(x) = e^∫P(x)dx

y= 1/I(x) (∫I(x)Q(x)dx +C)

Question 12

Homogenous ODE what they look like?

Answer 12

M(x,y) + N(x,y)dy/dx = 0

but M(x,y) and N(x,y) are both homogenous by n

which means M(λx,λy)= λ^nM(x,y)

Question 13

How to solve homogenous ODE?

Answer 13

sub in z(x) = y(x)/x

solve integral

get F(x,y)=0 or = c or however they ask

Question 14

Soln eq in terms of P(D)?

Answer 14

P(D-1)y=0 if 1 is a sol’n
alt P(D-1)(D+3) if 1 and =3 are sol’n

Question 15

Derivative of cos(x)?

Answer 15

-sinx

Question 16

1/x^2+1

Answer 16

arctan x

Question 17

1/x^2-1

Answer 17

arcsin x