Question

I need to find the partivular solution to the differential equation dy/dx=(1+y)/x Given the initial conditon f(-1)=1

Answer 1

I love diff equations! =)
So first of all see if its separable

Answer 2

it is!

Answer 3

So separate it:
\[dy/dx = (1+y)/x\]

Answer 4

Sorry, I keep crashing on here but i will always reply:
So multiply both sides by dx and divide both sides by 1+y

Answer 5

You will have:
\[dy/(1+y) = dx/x\]

Answer 6

can't you think of it as dy/(1+y) you're in good hands later.

Answer 7

Integrate both sides:
Left side will be just Ln|1+y| and the right side will be Ln|x|

Answer 8

So:
Ln|1+y| = Ln|x| + C

Answer 9

Raise e to both sides to get rid of Ln:
\[e^{Ln|1+y|} = e^{Ln|x|+C}\]

Answer 10

Rewrite and simplify:
\[e^{Ln|1+y|} = e^{Ln|x|}*e^C\]
\[1+y = Kx\]

Answer 11

(e to some constant is another constant so I just let that other constant be K)
Now apply initial conditions:
f(-1)=1:
1+1 = -1K; 2 = -K; K = -2

Answer 12

So your answer is:
1+y = -2x, or:
y = -2x - 1 <= Final answer

Answer 13

Im taking a differential equations course! IT IS FUN!!
P.S.:
Please click on become a fan if I helped, I really want to get to the next level!! Thanks =)

Answer 14

why do you multiply by e^c

Answer 15

there's a property:
\[A^{B+C} = A^B * A^C\], so in our case:
\[e^{Ln|x|+C} = e^{\ln|x|} * e^C\]

Answer 16

and e to some constant is another constant so I said let that constant be K. Any other questions?

Answer 17

no and thanks for your help