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Amisha F.

Calculus 1 / AB

1 week, 3 days ago

Solve the homogeneous differential equation. $y^{\prime}=\frac{x+y}{2 x}$

Chapter 6

Differential Equations

Section 3

Separation of Variables and the Logistic Equation

for this problem. We are asked to solve the homogeneous differential equation. Why prime equals X plus Y over two X. So the first thing that we can do here is rewrite this as being why prime equals X over two X. Or just 1/2 plus Y over two X. Which means that we can then rewrite this into the typical homogeneous form of Y. Prime. Actually I'll write it as Y prime plus negative 1/2 X times Y equals one half. So now that we have this in the typical or standard homogeneous form, we can find our integrating factor which I'm going to call if well that's going to be E to the power of the integral or negative of the integral of 1/2 X. So can actually simplify that a little bit further. That would be each the power of negative 1/2 times the integral of one over X. Dx. Which in turn means that this will be E. To the power of negative one half. Lawn of X. Yeah, so that means then using the rules of logarithms that that is in turn going to be each power of lawn of X. The power of negative 1/2. So that then means that are integrating factor lastly is going to be just X. Power of negative 1/2. So using that then we would have that the left hand side Is going to be x power of negative 1/2 times. Why prime Should equal 1/2 times x power of negative one half. So we integrate both sides. Which just means removing the prime on the left hand side. So we get exposure of negative one half. Why equals one half. Now the integral of X power of negative one half would be two X power of 1/2 plus C. So we have ah why over route X equals route X plus a constant. Then we multiply through everything by route X. So we get that Y is going to equal root X times route X. So just X squared plus some constant times the square root of X.

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