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Solve the differential equation.

$ 2xy' + y = 2 \sqrt x $

$$

\begin{array}{c}{y=\sqrt{x}+\frac{C}{\sqrt{x}}} \\ {\text { Hint: Integrating factor is } \sqrt{x}}\end{array}

$$

Differential Equations

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

Idaho State University

before we find the integrating factor, we should divide both sides by two acts in order to get into standard form. Why prime plus y times P of x is Q of axe doing this we have. Why Prime plus y divide by two of acts is one over square root off acts. So now on to the integrating factor, each the integral of one over to axe de acts can be written as each the natural lot of X to the 1/2 which is the same thing. A squirt of ex each. The natural log is one which means we have squared of acts or extra 1/2 is our integrating factor. OK, this works out because we now have squared of acts which is our integrating factor. Like I just said, being multiplied by all our terms so you can see I'm doing that right now. I'm multiplying this by all of our terms. Okay, Now that we have this, we know that we have essentially, we're gonna be integrating D of acts on the right hand side. We don't do anything to the left hand side. That remains is why Squared axe now integrating D of X of the same things. Interesting one interpreted. One simply gives us X When we take the integral, we also have to add our seat Constant of integration. Divide both sides by squirt of X to divide all these terms by scores of X, we get why escort of acts plus C divided by square root of X.