Question

Solve the differential equation (dy/dx)=ycos(x)/(1+y^2) with the initial condition y(0) = 1.

Answer 1

\[\frac{ dy }{dx }=\frac{ y \cos(x) }{ 1+y ^{2}}\]

Answer 2

First we separate the values but idk how to do that in this case.

Answer 3

@Michele_Laino

Answer 4

if we separate the varibles, we get:
\[dy\frac{{1 + {y^2}}}{y} = \cos x\;dx\]

Answer 5

variables*

Answer 6

@Michele_Laino then we intergrate can get this \[\int\limits \frac{ 1+y ^{2} }{ y } dy= \int\limits \cos(x) dx\]
\[\ln(y)+\frac{ y ^{2} }{ 2}+C=\sin(x)+C\]
\[\ln(y) + \frac{ y ^{2} }{ 2 }=\sin(x)+C\]

Answer 7

@Michele_Laino what then?

Answer 8

we have to set y=1 and x=0, so we can find the value of the constant C

Answer 9

okay so plug in the x and y values or should we simplify the rest of the equation first. can if we have to simplify what would i do next?

Answer 10

@Michele_Laino

Answer 11

it is the same, we can rewrite your solution as below:
\[\Large y{e^{{y^2}/2}} = C{e^{\sin x}}\]

Answer 12

substantially it is the same as yours

Answer 13

if I set x=0 and y=1, I get this:
\[\Large C = \sqrt e \]

Answer 14

Yes! I got the same answer so now what do we do? @Michele_Laino

Answer 15

plug sqrt(e) back into the equation

Answer 16

?

Answer 17

yes!

Answer 18

after that substitution, we get:
\[\Large y{e^{{y^2}/2}} = \sqrt e {e^{\sin x}}\]

Answer 19

\[ye ^{\frac{ y ^{2} }{ 2 }}=(\sqrt{e})e ^{\sin(x)}\]

Answer 20

that's right!

Answer 21

@Michele_Laino is that it then?

Answer 22

yes!

Answer 23

thanks!!

Answer 24

thanks! :)