Solving multivariable equation with integration

[gingermom]

Homework Statement

dy/dx = (1+x)/xy solve y(1) = -4

Homework Equations

The Attempt at a Solution

What threw me was the solve for if y(1) = -4

I grouped variables and then integrated both sides and solved for y.

(1/2)y^2 = ln|x|+x+c
y=+- √2ln|x|+x+c

I then switched the x and y to get the inverse of the equation and set that equal to -4

-4=-+ √2ln1+1+c
16= 2*(1+c)
8-1=c
7=c

However, I am not sure this was the correct approach. I could not figure out how to isolate x in the beginning, but I am not 100 percent sure of my logic of using the inverse, and whether I applied it appropriately. If this is way off base, can you point me in the correct direction? We have not had any problems quite like this and I am not sure I am making the correct leap in combining concepts.

Thanks.

 

[STEMucator]

Homework Statement

dy/dx = (1+x)/xy solve y(1) = -4

Homework Equations

The Attempt at a Solution

What threw me was the solve for if y(1) = -4

I grouped variables and then integrated both sides and solved for y.

(1/2)y^2 = ln|x|+x+c
y=+- √2ln|x|+x+c

I then switched the x and y to get the inverse of the equation and set that equal to -4

-4=-+ √2ln1+1+c
16= 2*(1+c)
8-1=c
7=c

However, I am not sure this was the correct approach. I could not figure out how to isolate x in the beginning, but I am not 100 percent sure of my logic of using the inverse, and whether I applied it appropriately. If this is way off base, can you point me in the correct direction? We have not had any problems quite like this and I am not sure I am making the correct leap in combining concepts.

Thanks.

Slight miscommunication here. Your second line should read:

$y= +- \sqrt{2ln|x| + 2x + 2c}$

Then $y(1) = -4$ would indeed imply $c = 7$.

Then simply subbing $c$ into your solution from prior would be the last thing to do.

 

[gingermom]

Thanks - Yes, sorry about that I had used parenthesis but they disappeared and I didn't notice. So using the inverse to find C was the right thing to do. Yeah, maybe I am getting this after all! I was so focused on finding C I forgot about putting it back in. Should have posted that part, too I guess. First time here. Will try and do better if ( I probably should say when) I need additional help. Thanks again.