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Problem 4 Medium Difficulty

(a) Show that every member of the family of functions $y=(\ln x+C) / x$ is a solution of the differential
equation $x^{2} y^{\prime}+x y=1$
(b) Illustrate part (a) by graphing several members of the family of solutions on a common screen.
(c) Find a solution of the differential equation that satisfies the initial condition $y(1)=2$ .
(d) Find a solution of the differential equation that satisfies the initial condition $y(2)=1$

Answer

$$
y=\frac{\ln \frac{x}{2}+2}{x}
$$

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Mshahidmshahidm S.

April 22, 2021

03085570113

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Mshahidmshahidm S.

April 22, 2021

Video Transcript

All right, So this is a four part question. If you must play both sides by acts, we end up with X. Why is equal to L a X plus seat differentiating respect act? We end up with X y prime plus why times once its wife is equal to one. So therefore we improved it. Let's party for part B if you were to grasp the values we end up with here of with several see values. So there several points to a graph and thes for other see values as so from see values of native 5 to 5. Okay, so for the next one, it is given that wise thing with Ellen of X plus seed about the extra general solution to find a solution that satisfies a bite of one easy to identify seat substituting and X is equal to one. And why you put the two unsolved for Syria. End up with two is equal to you, Ellen of +10 plus one. So therefore C is equal to two substituting back in the value of Syria and the wise people to Ellen of explosives to do about my ex. That's for part C. Finally have Part D. This is basically just a different solution with the substitute X equals issue. Unwise equal toe Want to end up with why is equal to Ellen of two plus C, divided to multiply and both sides by 200 soldiers perceiving that sees equal to minus Eleanor too, substituting back in the value of the values you would end up with l N L N off X plus two minus ln of two divided by X, divided by axe, which is equal to you ln of X divided by two plus two plus two divided by axe.