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clarification with simple differential equation

Hi all! My problem concerns the simple differential equation:

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

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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:

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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.)

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.

Re: clarification with simple differential equation

Absolutely right both the solutions are right.

Re: clarification with simple differential equation

Another approach would be to observe the characteristic root $\displaystyle r=0$ is of multiplicity 4, hence the general solution is:

$\displaystyle y(x)=c_1+c_2x+c_3x^2+c_4x^3$

Re: clarification with simple differential equation

russo's answer is correct. So your $\displaystyle c_1$ is 6 times the book's $\displaystyle c_4$, your $\displaystyle c_2$ is 2 times the book's $\displaystyle c_3$, your $\displaystyle c_3$ is the book's $\displaystyle c_2$, and your $\displaystyle c_4$ is the book's $\displaystyle c_1$.

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