# Thread: ODEs with a particular solution

1. ## ODEs with a particular solution

You are given that is a particular solution to the nonhomogeneous differential equation . Determine the general solution of the DE. Note: Any arbitrary constants used in the answer must be a lower-case "c".

So far I've found the general solution without the particular solution to be:

$\displaystyle y = t^3/5+c/t^2$

Now what do I do with the particular solution? Any help is appreciated!

2. Once You know a particular solution of the 'complete' DE, in order to solve the problem You have to find the general solution of the 'incomplete' DE that is...

$\displaystyle t\cdot y^{'} - 2\cdot y=0$ (1)

This linear DE is 'Euler type' and its solution has the form $\displaystyle y= c\cdot t^{\alpha}$. It is easy to verify that the only value of $\displaystyle \alpha$ that satisfies (1) is $\displaystyle \alpha=2$ so that the general solution of (1) is...

$\displaystyle y=c\cdot t^{2}$ (2)

... and the general solution of the 'complete' DE is...

$\displaystyle y=c\cdot t^{2} + t^{3}$ (3)

Kind regards

$\displaystyle \chi$ $\displaystyle \sigma$