# Differential equations - 2nd order euler eq'n

(a) Let alpha (a) and beta (b) be given constant. show that t^r is a solution of the Euler equation
t^2 d^2y/dt^2 + at dy/dt + by = 0 , t>0
if r^2 + (a-1)r + b = 0

(b) suppose that (a-t)^2 = 4b. Show that (ln t)t^(1-a)/2 is a second solution of Euler's equation.

for (a) it says, shown that t^r is a solution, so can i just substitute in ?

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siddharth
Homework Helper
Gold Member
for (a) it says, shown that t^r is a solution, so can i just substitute in ?
Indeed! If y=f(t) is a solution, then it satisfies the differential equation.

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HallsofIvy
Homework Helper
(a) Let alpha (a) and beta (b) be given constant. show that t^r is a solution of the Euler equation
t^2 d^2y/dt^2 + at dy/dt + by = 0 , t>0
if r^2 + (a-1)r + b = 0

(b) suppose that (a-t)^2 = 4b. Show that (ln t)t^(1-a)/2 is a second solution of Euler's equation.

for (a) it says, shown that t^r is a solution, so can i just substitute in ?
Yes, that's how you show that anything is a solution to any equation!

However, "suppose that (a-t)^2= 4b doesn't make sense- a and b are constants while t is a variable. Surely you mean "suppose that (a-1)^2= 4b", in which case the "characteristic equation" r2+ (a-1)r+ b= 0 has (1-a)/2 as a double root. Again, just let y= (ln t)t(1-a)/2 and show that that satisfies the equation in the special case (1-a)2= 4b.

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Am I doing something wrong.
I substituted y(t) = t^r in , and end up getting r^2 + (a-1)r + b = 0

here is my work:

y(t)=t^r
y'(t) = rt^(r-1)
y'(t) = r(r-1)t^(r-2)

t^2(r)(r-1)t^(r-2) + atrt^(r-1) + bt^r = 0
r(r-1)t^r + art^r + bt^r = 0
r(r-1) + ar + b = 0
r^2 - r + ar + b = 0
r^2 + (a-1)r + b = 0

i have no idea what to do now

HallsofIvy
Homework Helper
It might help to go back and read the question again! It said: ". show that t^r is a solution of the Euler equation
t^2 d^2y/dt^2 + at dy/dt + by = 0 , t>0
if r^2 + (a-1)r + b = 0"
Isn't that exactly what you just did? Replacing y by rt satisfies the equation (makes it equal to 0) if and only if your last equation r2+ (a-1)r+ b= 0.

Now do the same for tr2, remembering that you now have the additional requirement that "(a-1)2= 4b" (NOT (a-t)2!).

ohh, wow i totally forgot about the if r^2+(a-1)r + b=0 part, thanks.
but for (b), why "do the same for tr^2", why am i not subbing in the (ln t)t^1-a/2

can anyone help with this?
i have tried subbing in (ln t)t^1-a/2 but i do not get 0, so i know i am doing somethi gnwrong and it's not my work that's wrong as i've done it twice with the same answer. it keeps coming out to a^2 - 3 = 0

yo

braindead, do u by chance to queen's

HallsofIvy