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anonymous
 one year ago
don't know how to solve this differential equation
anonymous
 one year ago
don't know how to solve this differential equation

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freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[\frac{dy}{y}= 6x \sqrt{3x^21} dx\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay real quick wouldn't it be dy/dx on the left side?

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.2separation of variables.... in the form of h(y) dy = f(x) dx

freckles
 one year ago
Best ResponseYou've already chosen the best response.2i multiplied both sides by dx

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.2you need all y's to the left and all x's to the right... then integrate both sides..

freckles
 one year ago
Best ResponseYou've already chosen the best response.2i also divided both sides by y and multiplied both sides by 6x

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay i see one sec let me integrate

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay so im getting \[\frac{ 2(3x^2 1)^3/2 }{ 3}+C\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0not sure how to do the left side

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[\frac{2}{3}(3x^21)^\frac{3}{2}+C \text{ is \right for the \right hand side}\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.2the left hand hand is just lny+k but you only need one constant of integration in your equation

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[\lny=\frac{2}{3}(3x^21)^\frac{3}{2}+K\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0wait how did you get that from dy/y?

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[\frac{1}{6x} \frac{dy}{dx}=y \sqrt{3x^21} \\ \text{ divide both sides by } y \\ \frac{1}{6x} \frac{1}{y} \frac{dy}{dx} = \sqrt{3x^21} \\ \text{ multiply both sides by } 6x \\ \frac{1}{y} \frac{dy}{dx}= 6x \sqrt{3x^21} \\ \text{ multiply both sides by } dx \\ \frac{1}{y} dy= 6x \sqrt{3x^21} dx \\ \text{ or this can also be written as } \frac{dy}{y}=6x \sqrt{3x^21} dx\]

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.21/y dy the antiderivative of 1/y is just ln y

freckles
 one year ago
Best ResponseYou've already chosen the best response.2oh not how I got dy/y

freckles
 one year ago
Best ResponseYou've already chosen the best response.2but how you integrate it

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.2but we're still not done because we need a y by itself... we have to take e^ {} on everything so that we have y = (all the contents)

freckles
 one year ago
Best ResponseYou've already chosen the best response.2the derivative of lny w.r.t to y is y'/y=1/y

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0no i ment how did you integrate dy/y

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[\int\limits \frac{du}{u}=\lnu+K \text{ since } \frac{d}{du}(\lnu+K)=\frac{1}{u}+0\]

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.2first rewrite this as 1/y dy then the antiderivative of 1/y is ln y it's a logarithm the derivative is 1/y (1) because we also have to take the derivative of the denominator but it's just 1, so it's 1/y only

freckles
 one year ago
Best ResponseYou've already chosen the best response.2well actually it is a definition in most calculus books the following: \[\ln(x)=\int\limits_1^x \frac{1}{t} dt\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay i get it so now i just isolate y right?

freckles
 one year ago
Best ResponseYou've already chosen the best response.2you can if you are asked to find an explicit form for y

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0it just says solve the differential equation ... so i assume i just solve for y

freckles
 one year ago
Best ResponseYou've already chosen the best response.2you could stop at \[\lny=\frac{2}{3}(3x^21)^\frac{3}{2}+K\] then but solving this one for y is doable

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.2e^{ln y} e and ln are inverses of each other so it cancels out for y we're you given initial conditions in your problem ? if you were, then we can solve... if not we can just take e^{} on both sides

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay but then how do i get rid of the absolute value ?

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.2\[\large \lny=\frac{2}{3}(3x^21)^\frac{3}{2}+K\] \[\large e^{lny}=e^{\frac{2}{3}(3x^21)^\frac{3}{2}+K}\] \[\large y =e^{\frac{2}{3}(3x^21)^\frac{3}{2}+K}\]

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0these are my answer choices by the way and none of the above

freckles
 one year ago
Best ResponseYou've already chosen the best response.2as long as y is not 0 of course

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ugh i thought your K was your way of saying Constant

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.2forgot that part... e^K = C substitution.

freckles
 one year ago
Best ResponseYou've already chosen the best response.2\[\text{ assuming } y>0 \text{ we do have } \\ y=e^{\frac{2}{3}(3x^21)^\frac{3}{2}+K} \\ \\ \text{ but we can also rewrite this } \\ y=e^{k} e^{\frac{2}{3} (3x^21)^\frac{3}{2}} \text{ by law of exponents } \\ \text{ so now \it doesn't matter much about the } y>0 \text{ since } C \\ \text{ could be negative or positive } \\ \\ y=C e^{\frac{2}{3}(3x^21)^\frac{3}{2}}\]

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.2\[\large y =e^{\frac{2}{3}(3x^21)^\frac{3}{2}}e^{K} \] \[\large y =e^{\frac{2}{3}(3x^21)^\frac{3}{2}}C\]

freckles
 one year ago
Best ResponseYou've already chosen the best response.2@UsukiDoll is not really part you forgot. it is just a little something extra that isn't really needed but people write because it does look prettier

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay so wtf is K then

freckles
 one year ago
Best ResponseYou've already chosen the best response.2you are still replacing an unknown constant with another unknown constant

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0why did you use K instead of c then

freckles
 one year ago
Best ResponseYou've already chosen the best response.2you can mean any letter to be a constant

freckles
 one year ago
Best ResponseYou've already chosen the best response.2I used K as a constant above we ended up with e^K which still an unknown constant you can replace that with another unknown constant call it C

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.2you can have any letters as a constant tbh. but it's usually C

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay i understand that the e^anyunknown constant equals another constant my question was why you used two different symbols for constant in the same equation

freckles
 one year ago
Best ResponseYou've already chosen the best response.2e^K =C doesn't imply K=C

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.2I usually just leave the solution alone...unless it begs to be changed .

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0okay so then its just to keep track of which constant is which , okay that sort of makes sense

freckles
 one year ago
Best ResponseYou've already chosen the best response.2it seems funny to me to replace e^C with C

freckles
 one year ago
Best ResponseYou've already chosen the best response.2it is like replace e^5 with 5

freckles
 one year ago
Best ResponseYou've already chosen the best response.2even though e^5 doesn't equal 5

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0i find it soothing to just be able to take out constants like that

freckles
 one year ago
Best ResponseYou've already chosen the best response.2but yeah you could have just said e^C then replace it with C i think some people do that

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0anyway thank you very much for the help guys i wish i could give you both a medal :D

anonymous
 one year ago
Best ResponseYou've already chosen the best response.0ugh who do i give it too D:

UsukiDoll
 one year ago
Best ResponseYou've already chosen the best response.2I gave one to freckles already... so......?
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