Question

please help! question in comments

Answer 1

\[y(x)=2+\int\limits_{1}^{x}\left(\begin{matrix}dt \\ ty(t)\end{matrix}\right)\]

Answer 2

x>0

Answer 3

Did you try differentiating ?

Answer 4

\[\frac{ dy }{ dx }=\frac{ dx }{ xy(x) }\]

Answer 5

Is this part correct?

Answer 6

dx wont be there after differentiating

Answer 7

\[\frac{ dy }{ dx }=\frac{ 1 }{ xy }\]

Answer 8

separate variables and solve the DE

Answer 9

\[\frac{ y^2 }{ 2 }=\ln \left| x \right|+C\]

Answer 10

\[C=\frac{ y^2 }{ 2 }-\ln \left| x \right|\]

Answer 11

But what do I do from here?

Answer 12

use the intial condition to find C

Answer 13

y(1) = 2

Answer 14

plugin x = 1, y = 2 in the final equation

Answer 15

\[C=\frac{ 2^2 }{ 2 }-\ln \left| 1 \right|\]

Answer 16

So C=2

Answer 17

\[y=\sqrt{2\ln \left| x \right|+4}\]

Answer 18

Is this right?

Answer 19

Looks good!

Answer 20

Thank you!