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Find the complete solution of the linear differential equation dy y r' _ dr...

Question

Find the complete solution of the linear differential equation dy y r' _ dr

Find the complete solution of the linear differential equation dy y r' _ dr



Answers

Find the general solution to the linear differential equation.
$$
y^{\prime \prime}+y^{\prime}+y=0
$$

Okay. We'll start this problem by rewriting why prime is Dy Dx and this equals to why and then we can move terms around. So let's keep the wives to one side. So I. Y. Dy sorry one over. Why do I? It was one of the access to the other side. So have one over X. Dx. Okay integrate both sides. So on the left hand side we should integrate to natural log the absolute value of Y. On the right hand side we'll integrate to the natural log of the absolute value of X plus C. Now we raise both sides to eat power. Well have Y equals to see X. And that's our answer.

To start off this problem, we will want to divide off the two in front of the wide of abrupt. This will leave us with my double. Prime is equal to zero Now we will want to integrate both sides. These are indefinite integral. So be careful of the constant. The integral of why double prime will be wide prime plus a constant and the integral of zero will be zero plus a separate constant. These constants ca NBI added together and we can just have a new constant which we can label C two on the left side. Now we will want to integrate both sides again. When doing this, the integral of UAE prime will be why and the integral of C two will be C two. I'm sex now Will you will need to add on another constant. The integral of zero is zero plus a separate constant. Again these constants can again be combined into a different constant weaken label See one Now we will want to get why alone To do this, we will want to subtract off see two times X and C one This will leave us with y equals negative C two times X minus C. One. Because thes constants are unknown, the negatives can year raced. This is because when doing the algebra behind solving for them, the negatives would cancel out if it was a positive, and if the constant needed to be a negative, the constants would show negative, no matter what. Writing it, as if the constant were positive makes it easier to see what the equation is.

All right. Let's start off this problem by multiplying by Y and multiplying by X. R. D. X. On both sides. So we'll have is why do you why equals two X. D. X. And now to get rid of the Dy and dx, we can integrate both sides. So take the integral. And the inauguration of why is why squared divided by two and the integration of axes X squared divided by two plus C. Right? And we can multiply both sides by two. So I have Y squared equals two X squared plus C. And we don't have to put the two C. Because C um conglomerates any constance and you can basically ignore. Right? So that's your answer.


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