Question

Given the differential equation

(x^2+2x-1)y''-2(x+1)y'+2y=0

Show that the equation has a linear polynomial and a quadratic polynomial as solutions.

Answer 1

For linear:
start with y = mx+b
then
y' = m
y'' = 0
plug these into diff equ and simplify, if all the "x" terms cancel out then "y" can be linear under certain conditions

do same thing for quadratic:
start with y = ax^2 +bx +c
y' = 2ax +b
y'' = 2a

Answer 2

thank you very much. easy enough when explained. :-). so for linear I got m=b, and for quadratic i got c=a+b. I am then asked to come up with two linearly independent solutions of equation and give general solution.

so all i need to do is just let y=mx+b, and y=ax^2+bx+c, and set m=c=1, or any constant, and I get y=x+1 and y=x^2+x+2, and use both of those for my general solution....

does it matter what I set the constant = too.... could I of used y=2x+2 and y=2x^2+2x+4 instead?

Answer 3

no doesn't matter what constant you pick....for general solution , just recognize fact that if a=0 in quadratic solution you end up with linear solution

y = ax^2 +bx + (a+b)
a=0
y = bx + b

thus general solution is:
y = ax^2 + bx+ (a+b)

for unique solution, you would need 2 initial values to determine constants (a,b)