₦airaland Forum

Madmathecian:
Solve

Goodyshoes:
Lastly.....
Solve for x:

x^4 + x^5 = 12x^x

Please clear solutions.

Karmanaut:
Easy.
Set tan(x) as sin(x)/cos(x)
So you'll rewrite the question as:
lim^x->π/2 sinx^cosx/cosx^cosx
Then you substitute π/2 for x.
Since sinx^cosx= 1^0 =1 and
Since cosx^cosx resolves to 0^0 which is mathematically undefined (0 or 1, depending on who you ask) substitute y for cosx.
So you have 0/y^y
Then using the exponential function the denominator becomes e^{lny^y}
which is e^y*lny
Since y = cos π/2 = 0
we have:
e⁰
which is 1.
So we have 1/1
= 1.
Done.

ladokuntlad:
Boss u got the answer but ur approach got error at bolded part
u will get maths error(undefine) if u insert 0 in place of y for lny

So will say L'HOPITAL's rule is stil d genuine way out until u proof otherwise

Karmanaut:
It's actually correct, you don't evaluate it directly. (ln(0) is undefined)
Instead I used the standard limit:
lim_y->0⁺ y*ln(y) = 0.
So instead of evaluating e^y*ln(y)
You replace it with 0
So that it becomes e⁰
= 1.

ladokuntlad:
Guys i av some maths Olympiad questions for u to solve

ladokuntlad:
House i av a cowbell question to give

x+y+z=1
x²+y²+z²=35
x³+y³+z³=97

Show workings if x=-1 y=-3 and z=-3

ladokuntlad:
House i av a cowbell question to give

x+y+z=1
x²+y²+z²=35
x³+y³+z³=97

Show workings if x=-1 y=-3 and z=-3

ladokuntlad:
APART FROM GRAPHICAL METHOD WICH CLEARLY SHOWS THAT THE ANSWER IS 2 or 3
We can also use NEWTON-RAPHSON method i.e
[img]http://web.mit.edu/10.001/Web/Course_Notes/NLAE/equation7.gif[/img]
where X_i denotes first guess or initial number
F(X_i) denotes d solution at X_i
and F'(X_i) denote solution of the derivative at X_i
F'(X_i)= 4x³+5x⁴-12x^x(lnX+1)
Now let my first guess be 2( u can use any number of ur choice)
F'(2) 30.73 and F(2)=0

inserting into the formula gives
X₁=2-(0/30.73)=2
its obvious we going to have a reoccurring solution
thus , x=2.

Also doing the same for X_i=3 gives same approach.
we conclude that x=2 or x=3

ladokuntlad:
APART FROM GRAPHICAL METHOD WICH CLEARLY SHOWS THAT THE ANSWER IS 2 or 3
We can also use NEWTON-RAPHSON method i.e
[img]http://web.mit.edu/10.001/Web/Course_Notes/NLAE/equation7.gif[/img]
where X_i denotes first guess or initial number
F(X_i) denotes d solution at X_i
and F'(X_i) denote solution of the derivative at X_i
F'(X_i)= 4x³+5x⁴-12x^x(lnX+1)
Now let my first guess be 2( u can use any number of ur choice)
F'(2) 30.73 and F(2)=0

inserting into the formula gives
X₁=2-(0/30.73)=2
its obvious we going to have a reoccurring solution
thus , x=2.

Also doing the same for X_i=3 gives same approach.
we conclude that x=2 or x=3

MathsChic:
See below attachments for the solution. Typically would have loved to type them out here with html, but it'll be cumbersome. Don't know why this site is like that

But a solution can't exist at x=0. I've explained all the details. Please bear with my almost illegible handwriting

Solving normally...
xy'-y-x-1=0
y'-y/x =1+1/x
(x^-1y)'=1/x + 1/x²
(x^-1y) = Inx - 1/x
y=xInx - 1

The power series solution is simply an approximation of the solution curve at x=1, which should give a good solution at that point. Should be desire a solution at any other point, we can do x = x_o. We can't get a solution at x=0 because of the Inx curve. And the first order DE showed it in the coefficients of y' and y, since 1/x is not analytic at x=0.

MathsChic:
I like this.

agentofchange1:
see mathematicians abeg ..
greetings. all .

jackpot:
Hey agentofchange

MathsChic:
That's the solution above, but ama try expatiate.
∫(sinx)²dx

We know that cos(2x) = 1 - 2(sinx)²
So

∫(1 - cos(2x))/2dx
∫(1/2 - cos(2x)/2)dx

=(1/2)x - sin(2x)/4 + c
Since ∫(cosx)dx = sinx

Therefore
=(1/2)(x - sin(2x)/2) + c

Sin(2x) itself is 2sinxcosx

Therefore
= (1/2)(x - sinx.cosx)

agentofchange1:
hey , buddy , howdy ?

Karmanaut:
The question has been asked and answered in this thread
Cheers.

agentofchange1:
hey , buddy , howdy ?

Karmanaut :
.

ladokuntlad:
Boss u av some of my questions to answer naaa. Oya kindly giv me solutions to it

agentofchange1:
where ?

ladokuntlad:
Scroll up maths olympiad questions

agentofchange1:
oh , really ? like for. how much are we talking about here ?

ladokuntlad:
Go meet d Australians dat set d questions. Dey ar secondary school maths oooo