Differential Models

Question 1

if you have an input force f1(t) that produces output displacement x1(t), and similarly for f2(t) and x2(t), how would you superimpose them

Answer 1

af1(t) + bf2(t) = ax1(t) + bx2(t)

Question 2

what are the 4 steps in solving linear system model problems

Answer 2

• draw a diagram of a small element
• balance forces or energy fluxes to get an ordinary diff eqn
• take the deltas to 0 to get an ODE
• solve the ODE

Question 3

for linear diff eqns like y(n) - 2.95y(n-1) + 2y(n-2) = 0, how would you start solving it

Answer 3

• youd need to write it in term of lambda to get a quad eqn
• in thise case its Y^2 - 2.95Y + 2 = 0
• its hard to explain just looks at the notes

Question 4

if f is a function of x and y f(x,y) how would you partially differentiate it

Answer 4

• df/dx = diff of x, treating y as a constant
• df/dy = diff of y, treating x as a constant
• the ‘d’ in this case is actually a phi

Question 5

what would be partial derivatives of f(x,y) 2yx^3 + 2xy^3

Answer 5

• df/dx = 6y*x^2 + 2y^3

- df/dy = 2x^3 + 6x*y^2

Question 6

you want to know the gradient along a particular direction u = (a,b) where u is a unit vector. the rate of change of f going in this direction is Du f(x,y) = lim as h tends to 0 (idrk what h is). what is the formula for figuring it out

Answer 6

a(df/dx) + b(df/dy)

Question 7

if you had f(u(x,y)), how would you apply the chain rule to work out df/dx and df/dy

Answer 7

df/dx = df/du * du/dx, likewise for y

Question 8

what if you have f(u(x,y), v(x,y))

Answer 8

youd do the same thing for v then add the results to gether