# Linear and homogeneous ODE's

• August 22nd 2008, 08:40 AM
Linear and homogeneous ODE's
I have the following two ODE's
$\frac{d^2y}{dx^2} = -p^2y+1
$

$\frac{d^2y}{dx^2} = -(p+1)^2y$
And am asked which is linear and which is homogeneous.
Then I must find the general solution for the second and the particular solution with dy/dx=0, y=1 when x=0

Any help is greatly appreciated.
• August 22nd 2008, 09:05 AM
Jhevon
Quote:

I have the following two ODE's
$\frac{d^2y}{dx^2} = -p^2y+1
$

$\frac{d^2y}{dx^2} = -(p+1)^2y$
And am asked which is linear and which is homogeneous.

this is the sort of thing you should look up in your text book.

for second order DEs:

Homogeneous means it looks like $p(x)y'' + q(x)y' + r(x)y = 0$ ......the 0 is what makes it homogeneous, in general, there would be a non-zero function on the right hand side.

Linear means we have a differential equation of the form $F(x,y,y', \dots, y^{(n)}) = 0$ in which the variables are the y-components, and all these components stand alone. meaning you only have things like y, y', etc and not things like yy' or y'y'' as the variables.

Quote:

Then I must find the general solution for the second and the particular solution with dy/dx=0, y=1 when x=0
for the first one (we need the method of undetermined coefficients), see here. for the second one, see here

hope that helps

it would be a good idea to look up linear and non-linear ODEs, homogeneous ODEs, non-homogeneous ODEs and second-order ODEs in your textbook. just look those up in the index. short of doing the problems for you (the majority of the second problem is done in one of the links i gave you, i think) it would take a lot of re-inventing the wheel to explain this. read your text and if you have specific questions, we can help
• August 22nd 2008, 10:35 AM
shawsend
Quote:

$\frac{d^2y}{dx^2} = -p^2y+1
$\frac{d^2y}{dx^2} = -(p+1)^2y$
Hey panda, should specify what $p$ is: constant, function of x or what it's generally meant to be in differential equations: $p=\frac{dy}{dx}$. The last is probably not what you have in mind but that one would make a nice challenging problem for those in here who like differential equations.