Question

Solve the separable differential equation
11 x - 6 y √(x²+ 1) (dy/dx) = 0.
Subject to the initial condition: y(0) = 4.
y =

Answer 1

rearrange to the form

f(y) dy = g(x) dx

Answer 2

can you do that @eescobedo1 ?

Answer 3

good , now integrate both sides

∫-6ydy =∫ -11/√(x²+1) (dx)

Answer 4

ok what I got was \[-3y ^{2}=-11\sqrt{x^{2}+1} + c\]

Answer 5

now solve it for y(x),

Answer 6

do you mean to use the initial condition at this point where I would let x=0 and y=4 to solve for c then substitute it back to the equation to solve for y?

Answer 7

yeah but i usually get the equation into y(x) = ... form before subbing in initlal condition

Answer 8

for this equation though it might be more sensible to just get it into y^2(x) = . . ., because you dont want to forget that extra solution

Answer 9

ok well then this is where I'm getting lost....

Answer 10

what do you get for

y^2

Answer 11

what I'm thinking is to divide the -3 by both sides to get \[y ^{2}=11\sqrt{x ^{2}+1}+c/3\]

Answer 12

yeah so
the result is

\[y^2(x)=\frac{11\sqrt{x^2+1}+c}3\]

if we change the constant of integration a bit \(C=c/3\)

\[y^2(x)=\tfrac{11}3\sqrt{x^2+1}+C\]

Answer 13

Now we can sub in y(0) = 4, ie x=0, y=4

Answer 14

and solve for C (or c)

Answer 15

ok let me try =)

Answer 16

actually i see now that \(C\) is not a good idea, it is simpler calculation with \(c\),

but they will give the same equation eventually

Answer 17

ok so I got c=-11, from this point I would sub my answer to the equation: \[y ^{2}=11/3\sqrt{x ^{2}+1}-11\]

Answer 18

that would be right, but i think there is a calculation error

Answer 19

\[16 \neq \tfrac{11}3 - 11\]

Answer 20

ok because the first time I worked it out I got c=-37 so at this point I would have \[y ^{2}=11/3\sqrt{x ^{2}+1}+37/3\]

Answer 21

yeah that's right \[y ^{2}(x)=\frac{11\sqrt{x ^{2}+1}+37}3\]

Answer 22

and we can check the initial point works

4^2 = (11+37) /3 = 16

it does!

Answer 23

if you really wanted to, you could get

\[y(x) = \pm\sqrt{\dots}\]

but
\[y^2(x) = \dots \] is much nicer

Answer 24

I know because we have to square both sides in or to get y, which is what they are asking, it just looks messy being that there is a square root over another square root...

Answer 25

yep, y(x) does look messy, even if you write the sqrts as ^{1/2}, not much you can do about it,
that why y^2(x)= ... is nicer

Answer 26

or maybe even \[3y^2(x) = 11\sqrt{x^2+1}+ 37\] is nicier still

Answer 27

Thank you for your help, pretty much I entered the answer as \[\pm \sqrt{....}\] and received an incorrect response of: Your answer isn't a formula that returns a number (it looks like a formula that returns a list of numbers

Answer 28

hmm, i would try entering y^2(x) = . . .

Answer 29

Ok after many attempts to input the answer correctly I finally was able to input it in the correct form and yes our answer was correct! I was just having trouble entering it in the program I am using, much thanks with helping me!!!

Answer 30

oh wait

if you check the detail

\[y ^{2}(x)=\frac{11\sqrt{x ^{2}+1}+37}3\\
y(x) = \pm\sqrt{\frac{11\sqrt{x ^{2}+1}+37}3}\]

remember
\[y(0) =4 \]

the negative solution dosen't fit the initial condition

Answer 31

because the sqrt term must be positive,

Answer 32

and you are correct I didn't submit the negative answer! =)

Answer 33

the plus/minus should only be there before we find c

Answer 34

ok, well i have to go,

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