Question

Solve the differential dy/dx=(1+x)/xy for x>0 and y(1)=-4. Thanks!

Answer 1

This can be done be separating the variables

Answer 2

Write as
g(y) dy=f(x) dx
and then integrate both sides

Answer 3

I have tried but failed.Can youplz howme th steps

Answer 4

I appeciate it

Answer 5

Well do you have
\[\frac{dy}{dx}=\frac{1+x}{xy} ?\]

Answer 6

yes

Answer 7

Ok did you multiply dx on both sides?

Answer 8

I got (x+1)/x dx=y dy

Answer 9

therefore 1+ (1/x) dx=y dy

Answer 10

then I integrated both sides

Answer 11

I got x+ lnx=(y^2)/2

Answer 12

(1+1/x) dx=y dy
yes
and yes
don't forget you need +C on one of those sides

Answer 13

yes

Answer 14

so great now use the initial condition y(1)=-4 to find C

Answer 15

can you me the rest

Answer 16

show*

Answer 17

You have this :
\[x+\ln(x)+C=\frac{y^2}{2}\]
Enter in 1 for x
Enter in -4 for y
Solve for C

Answer 18

-_> c=7

Answer 19

\[1+\ln(1)+C=\frac{(-4)^2}{2}\]
I just replaced x with 1 and y with -4 since the initial condition says y(1)=-4
\[1+0+C=\frac{16}{2}\]
\[1+C=8\]
Yes C=7 very good

Answer 20

I got y= -+ radical (2x+14+2lnx)

Answer 21

is tht right?

Answer 22

Yep gj! :) You multiplied 2 on both sides giving you
\[2x+2\ln(x)+2C=y^2\]
And we found C to be 7
So we have
\[y^2=2x+2\ln(x)+14\]
yes you are right :)

Answer 23

Thank you so much!!!