# clarification with simple differential equation

• January 4th 2013, 04:23 PM
zachd77
clarification with simple differential equation
Hi all! My problem concerns the simple differential equation:

Attachment 26467

I integrated four times and I ended up with this general solution (c constant):

Attachment 26468

I'd assume this is the right solution. I looked up the answer afterwards and I found that the textbook wrote the solution as follows:

Attachment 26469

Are these solutions the same? How would I simplify my answer to get the textbook's answer? Just looking to clarify my solution here.
(Not sure if this belongs in the Calculus forum or the Differential Equations forum. I apologize greatly if I chose the wrong forum to post this question in.)
• January 4th 2013, 05:54 PM
russo
Re: clarification with simple differential equation
As far as I know, both answers are right. Since 6 and 2 are constants, c4 and c3 "absorb" them. Then when you find a particular solution, it doesn't matter the way it's written, the constants will adequate to the right result. For example, let's say you find your first constant from your solution and it's 6, then you'll get 6/6 * x^3. If you find the constants using the textbook solution you'll find that the first constant is equal to 1.
• January 4th 2013, 11:15 PM
ibdutt
Re: clarification with simple differential equation
Absolutely right both the solutions are right.
• January 4th 2013, 11:24 PM
MarkFL
Re: clarification with simple differential equation
Another approach would be to observe the characteristic root $r=0$ is of multiplicity 4, hence the general solution is:

$y(x)=c_1+c_2x+c_3x^2+c_4x^3$
• January 4th 2013, 11:27 PM
hollywood
Re: clarification with simple differential equation
russo's answer is correct. So your $c_1$ is 6 times the book's $c_4$, your $c_2$ is 2 times the book's $c_3$, your $c_3$ is the book's $c_2$, and your $c_4$ is the book's $c_1$.

If you had additional information, like maybe initial values or boundary values, you would figure out what all the constants are, and the two general solutions would lead to the same specific solution.

ibdutt and MarkFL2 are also correct, of course. I didn't see their posts until after I posted mine.

- Hollywood