2nd Order ODE

Question 1

p(t) y′′+q(t) y′+r(t)y=0

Answer 1

2nd order linear equation that is homogeneous

Question 2

given ay’’ + by’ + cy = 0

and b^2 - 4ac > 0

Answer 2

general solution is y(t)=C1e^(r1t)+C2e^(r2t)

Question 3

given ay’’ + by’ + cy = 0

and b^2 - 4ac = 0

Answer 3

y(t)=C1e^(rt)+C2te^(r*t)

and r = -b/2a

Question 4

given ay’’ + by’ + cy = 0

and b^2 - 4ac < 0

Answer 4

y(t)=C1e^(λt)cos⁡(μt)+C2e^(λt)sin⁡(μt)
where λ = -b/2a and
μ = √(−(b^2−4ac))/2a

Question 5

The Wronskian of 2 functions f and g

Answer 5

fg’ - gf’

Question 6

Assume that y1 and y2 are solutions to some equation y(t) = ay’’ + by’ + cy
When are these solutions fundamental?

Answer 6

when the Wronskian of these solutions ≠ 0

Question 7

What does it mean for solutions to be fundamental?

What can we do with them if they are fundamental

Answer 7

We can add them together to form all possible solutions for the differential eqn. Ex//
y = C1y1 + C2y2

Question 8

t^2 y^′′+aty^′+by=0, t>0

What type of equation is this?

Answer 8

This is an Euler equation

Question 9

How do we solve this equation?

t^2 y^′′+aty^′+by=0, t>0

Answer 9

1) Set x=ln⁡(t)
2) The equation now becomes y’’+(a−1)y’+by=0
the y’s depend on x, NOT t
3) Solve this equation (Via the characteristic equation) and change back to t at the end

Question 10

what is the reduction of order formula?
(assume y’’ + p(t)y’+q(t)y = 0
and y1 is given)(solve for y2)

Answer 10

y2 = ( ∫ (e^(−∫p(t)dt)) /(y1^2) dt)) y1

The above solution is also fundamental

Question 11

p(t) y′′+q(t) y′+r(t)y=g(t)

Answer 11

2nd order linear equation that is nonhomogeneous

Question 12

variation of parameters method of finding the particular solution of nonhomogeneous diff eqs

Answer 12

Y = u1y1 + u2y2
where
u1 = - ∫(y2*g(t))/W(y1,y2)dt)
u2 = ∫((y1*g(t))/W(y1,y2)dt)
where W is the Wronskian

Question 13

Undetermined coefficients method of finding the particular solution of nonhomogeneous diff eqs

Answer 13

must have constant coefficients. g(t) needs to be
1) a polynomial
2) an exponential
3) a sine/cosine function
4) a combo of the above
Make your guess according to one of the above
(e.g. Y = a0 +a1t +a2t^2 or de^ct or a*cos(bt))
plug in to the original equation (Y -> y, Y’ -> y’ etc)
solve for the undetermined coefficients)

Question 14

How to tell beforehand whether my guess will work and what to do if it doesn’t. (Used in undetermined coefficients method)

Answer 14

1) Write homogeneous equation and find fundamental solutions y1 and y2.
2) Depending on the form of g, write a guess Y for a particular solution of the homogeneous equation
3) Check whether y1 or y2 (perhaps ,multiplied by an undetermined constant) appears in Y.
i) If So, multiply your entire guess by t to obtain a new guess. Repeat Step 3
ii) If Not, your guess is good

Question 15

How can you solve this?

p(t)y’’ + q(t)y’ + r(t) = g(t) + f(t)

Answer 15

Them up into two different NH eqns
(e.g. p(t)y’’ + q(t)y’ + r(t) = g(t) and p(t)y’’ + q(t)y’ + r(t) = f(t))
find the particular solution to both of these
(e.g. Y1 and Y2 respectively)
add these two up to find the particular solution of the given eqn
(e.g. Y = Y1 + Y2)

Question 16

What is the laplace transform of a function of f

or L{f(t)}

Answer 16

F(s) = ∫from 0 to ∞ of (e^(−st) f(t)dt)

Question 17

Laplace transform property of Linearity

Answer 17

L{cf(t)+dg(t)}=cL{f(t)} + dL{g(t)}
and for a number a
L{af(t)} = aF(s)
Careful: It is false that L{f(t)g(t)}=L{f(t)}•L{g(t)}

Question 18

Laplace transform property of Uniqueness

Answer 18

If L{f(t)}=L{g(t)} Then f(t)=g(t)
also if L{f(t)} = F(s) then
L^-1{F(s)} = f(t)

Question 19

L{f’(t)} and L{f’‘(t)}

Answer 19

L{f′(t)}=sL{f(t)}−f(0)

L{f′′(t)}=s^2 L{f(t)}−sf(0)−f′(0)

Question 20

given p(t)y’’ + q(t)y’ + r(t) = g(t) and g(t) = sin(3t) what should your guess be? (Undetermined coefficients)

Answer 20

Y = Asin(3t) + Bcos(3t)