Question

Solve the differential equation dy/dx=14x^2y^2 with the condition y(2)=4

the solution to the equation is y=

Answer 1

when given a seperable equation .... seperate it

Answer 2

how do we separate it?

Answer 3

\[\frac {dy}{dx}=f(x)~g(y)\]
\[\frac {dy}{g(y)}=f(x)~dx\]

Answer 4

its basic algebra to seperate it

Answer 5

so itll by dy/y^2=14x^2

Answer 6

**14x^2dx

Answer 7

of course: or

y^(-2) dy = 14x^2 dx

Answer 8

separate the variables and integrate.
substitute x=2,y=4 and find the value of constant and then substitute the value of constant.

Answer 9

im lost..

Answer 10

i cant see your work to determine why you are lost

Answer 11

or we suppose to find the derrivitve?

Answer 12

no, the integration

Answer 13

its already in derivative form

Answer 14

so itll be y^-1/-1 dy=14x^3/3 dx?

Answer 15

the dy dx parts are actually part of the operator and disappear

dont forget a +C at the end

Answer 16

so y^-1/-1=14x^3/3+c

Answer 17

good, now x=2 and y=4, solve for c

Answer 18

c=-451/12?

Answer 19

\[\frac 1y=\frac{14}3x^3+C\]
\[\frac 14=\frac{14}32^3+C\]
\[\frac 14-\frac{14}32^3=C\]

Answer 20

then rewrite for y in a more normal format of y= .....

Answer 21

-445/12 i believe

Answer 22

opps, i forgot the negative onthe y part

Answer 23

your right

Answer 24

how do we rewrite it?

Answer 25

1/y flips to y

just place a 1 over the right side to flip it

Answer 26

\[-\frac1y=\frac{14}3x^3-\frac{451}{12}\]
\[\frac1y=\frac{451}{12}-\frac{14}3x^3\]
\[y=\frac{1}{\frac{451}{12}-\frac{14}3x^3}\]
and simplify to your hearts content

Answer 27

so what do we do next

Answer 28

thats it, youve found "y" that fits the information

Answer 29

thanks!
can you help me with another problem similar to this

Answer 30

dy/dx=4x^4y^4 Y(2)=2

Answer 31

its not similar, its the exact same thing ...

Answer 32

seperate, integrate, and solve for C