# Two (probably stupidly simple) differential equations questions

• Oct 19th 2009, 01:39 PM
nathron
Two (probably stupidly simple) differential equations questions
Hello!

I never thought that there would be a math forum that would be this busy! Bookmarked.

Ok... enough with the sugar. (Wink) I have finished almost all of my homework sheet, but I am having problems with a couple questions. My text (First course in DE, by Zill) is not helping me at all, and neither is the almighty google.

The first is asking me to find a differential equation that is satisfied by the function $\displaystyle y=(C1+C2x)e^x$

I'm guessing it has to do with superposition, but I only know how to use that to get a solution from a DE, not vice versa!

Next up, I have a question asking me to find the intervals where these DEs have a unique solution:

a) $\displaystyle (x^2-1)y''+xy'-5y=3$
and
b) $\displaystyle (x^3-1)y''+3y=5$

EVERY question I've seen for finding intervals with a unique solution atleast has initial values.

I have been (Headbang), and of course, the math help centre at my school only has a DE assistant when I am at work, so any help you guys give me would be frickin awesome.

Thanks!
• Oct 20th 2009, 01:13 AM
chisigma
These are not at all stupid questions!...

First answer: the general solution of the DE is...

$\displaystyle y= c_{1}\cdot e^{x} + c_{2}\cdot x\cdot e^{x}$ (1)

... and that means that is a second order linear DE with constant coefficients and its 'characteristic equation' has solution $\displaystyle d=1$ with molteplicity two, i.e. is...

$\displaystyle d^{2} - 2\cdot d + 1=0$ (2)

The corresponding DE is...

$\displaystyle y^{''} -2\cdot y^{'} + y =0$ (3)

Second answer: the interval in which a linear second order DE has one and only one solution for any 'initial condition' is the inteval in which the 'coefficient' $\displaystyle a(x)$ of the term $\displaystyle y^{''}$ doesn't vanish, i.e. $\displaystyle a(x)\ne 0$. In a) is $\displaystyle a(x)= x^{2}-1$, in b) is $\displaystyle a(x)= x^{3}-1$, so that...

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$
• Oct 20th 2009, 03:39 PM
nathron
Thanks alot! That helped a lot. I can't believe how easy the second question was. It was worth as many marks as a question that took 5 times as long to do, so that is why I got tripped up by it.

:)

+1 Thanks!