Solving this differential equation

Gold Member
Homework Statement:
##(3x+4y)dx + (4x-3y)dy=0##
Relevant Equations:
##\frac{dy}{dx}##= ##v##+##x\frac {dv}{dx}##
##-\frac {dy}{dx}=\frac {3+4v}{4-3v}##
##\frac {3+4v}{4-3v}=-v-x\frac {dv}{dx}##
##-\frac {dx}{x}=\frac {4-3v}{8v-3v^2+3}##
##\frac {dx}{x}=\frac {4-3v}{3v^2-8v-3}##=[A/3v+1]+[B/v-3]##

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Gold Member
Homework Statement:: ##(3x+4y)dx + (4x-3y)dy=0##
Relevant Equations:: ##\frac{dy}{dx}##= ##v##+##x\frac {dv}{dx}##

##-\frac {dy}{dx}=\frac {3+4v}{4-3v}##
##\frac {3+4v}{4-3v}=-v-x\frac {dv}{dx}##
##-\frac {dx}{x}=\frac {4-3v}{8v-3v^2+3}##
##\frac {dx}{x}=\frac {4-3v}{3v^2-8v-3}##=[A/3v+1]+[B/v-3]##

ok, are my steps correct? then i can proceed. Actually i have a solution that is different from textbook, that's why i need a second perspective from you...i have ##A=-1.5## and ##B=-0.5##

Fred Wright
You have
$$\int \frac{\frac{4}{3}-v}{v^2-\frac{8}{3}v-1}dv$$
what are the roots of ##v^2-\frac{8}{3}v-1##?

Mentor
Homework Statement:: ##(3x+4y)dx + (4x-3y)dy=0##
Relevant Equations:: ##\frac{dy}{dx}##= ##v##+##x\frac {dv}{dx}##

##-\frac {dy}{dx}=\frac {3+4v}{4-3v}##
##\frac {3+4v}{4-3v}=-v-x\frac {dv}{dx}##
How did you get the line above? I would guess that you are doing something related to your Relevant Equations.
If so, the assumption there is that y = vx, so dy = vdx + xdv, and hence ##\frac{dy}{dx} = v + x\frac{dv}{dx}##.
Please show what you are substituting for v, and include the work that leads to the line below.
chwala said:
##-\frac {dx}{x}=\frac {4-3v}{8v-3v^2+3}##
##\frac {dx}{x}=\frac {4-3v}{3v^2-8v-3}##=[A/3v+1]+[B/v-3]##
Also, in the line above, you need more parentheses.
A/3v + 1 should be written as A/(3v + 1), and B/v-3 should be written as B/(v-3). The brackets you included are doing nothing to fix the problem - they need to be around the denominators, not the whole fraction.

DaveE and berkeman
Mentor
Isn't your original relationship an exact differential: $$3xdx+4d(xy)-3ydy=0$$

chwala
Gold Member
Thanks guys...let me respond later I finish with some errands, I will show my steps as asked...I didn't check to establish if the equation was exact, ...I will check that also. Regards

Gold Member
How did you get the line above? I would guess that you are doing something related to your Relevant Equations.
If so, the assumption there is that y = vx, so dy = vdx + xdv, and hence ##\frac{dy}{dx} = v + x\frac{dv}{dx}##.
Please show what you are substituting for v, and include the work that leads to the line below.
Also, in the line above, you need more parentheses.
A/3v + 1 should be written as A/(3v + 1), and B/v-3 should be written as B/(v-3). The brackets you included are doing nothing to fix the problem - they need to be around the denominators, not the whole fraction.

I will fix that too, in my subsequent working...

Gold Member
How did you get the line above? I would guess that you are doing something related to your Relevant Equations.
If so, the assumption there is that y = vx, so dy = vdx + xdv, and hence ##\frac{dy}{dx} = v + x\frac{dv}{dx}##.
Please show what you are substituting for v, and include the work that leads to the line below.
Also, in the line above, you need more parentheses.
A/3v + 1 should be written as A/(3v + 1), and B/v-3 should be written as B/(v-3). The brackets you included are doing nothing to fix the problem - they need to be around the denominators, not the whole fraction.

ok this was my approach,

##(3x +4y)dx +(4x-3y)dy=0##

##(3x +4y)dx =-(4x-3y)dy##

##\frac {3x+4y}{4x-3y}=-\frac {dy}{dx}##

Using, ##y=vx##

→##\frac {3x+4vx}{4x-3vx}=-\frac {dy}{dx}##

→##\frac {3+4v}{4-3v}=-\frac {dy}{dx}##

→##\frac {3+4v}{4-3v}=-v-x\frac {dv}{dx}##

→##-\frac {dx}{x}=\frac {4-3v}{8v-3v^2+3}##

→##\frac {dx}{x}=-\frac {4-3v}{8v-3v^2+3}##

→##\frac {dx}{x}=\frac {4-3v}{-8v+3v^2-3}##

does this negative also affect the numerator or is my working just fine?...

Gold Member
Isn't your original relationship an exact differential: $$3xdx+4d(xy)-3ydy=0$$
i will check it out, i have re posted my earlier approach though...

Mentor
i have A=−1.5 and B=−0.5
I get the same values.

ok this was my approach,
##(3x +4y)dx +(4x-3y)dy=0##
##(3x +4y)dx =-(4x-3y)dy##
##\frac {3x+4y}{4x-3y}=-\frac {dy}{dx}##
Using, ##y=vx##
→##\frac {3x+4vx}{4x-3vx}=-\frac {dy}{dx}##
→##\frac {3+4v}{4-3v}=-\frac {dy}{dx}##
→##\frac {3+4v}{4-3v}=-v-x\frac {dv}{dx}##
→##-\frac {dx}{x}=\frac {4-3v}{8v-3v^2+3}##
Above, you're missing dv from the right hand side.
chwala said:
→##\frac {dx}{x}=-\frac {4-3v}{8v-3v^2+3}##
→##\frac {dx}{x}=\frac {4-3v}{-8v+3v^2-3}##

does this negative also affect the numerator or is my working just fine?...
No, the negative affects only the numerator or only the denominator, but not both. If it were to effect both, that would be equivalent to multiplying by ##\frac {−1}{−1}##, which is different from having a factor of -1.
In your last line, you should add the dv factor, and write the denominator like this:
##\frac {dx}{x}=\frac {4-3v}{3v^2 -8v -3}dv##
Now rewrite the right side using your partial fraction decomposition, like so:
##\frac {dx}{x}=\frac {−3/2}{3v+1}dv+ \frac{−1/2}{v−3}dv##
Integrate both sides to get x in terms of v. Undo the substitution to get y back to end up with an equation that involves x and y.

@Chestermiller's suggestion was probably a lot simpler than what you're doing, but you've done most of the work already.

chwala
Gold Member
I get the same values.

Above, you're missing dv from the right hand side.
No, the negative affects only the numerator or only the denominator, but not both. If it were to effect both, that would be equivalent to multiplying by ##\frac {−1}{−1}##, which is different from having a factor of -1.
In your last line, you should add the dv factor, and write the denominator like this:
##\frac {dx}{x}=\frac {4-3v}{3v^2 -8v -3}dv##
Now rewrite the right side using your partial fraction decomposition, like so:
##\frac {dx}{x}=\frac {−3/2}{3v+1}dv+ \frac{−1/2}{v−3}dv##
Integrate both sides to get x in terms of v. Undo the substitution to get y back to end up with an equation that involves x and y.

@Chestermiller's suggestion was probably a lot simpler than what you're doing, but you've done most of the work already.

aaaaaaaaaargh, i really hate making those kind of mistakes...of leaving out the ##dv##. bingo mark, let me check it out...i will also use Chestermiller's approach and see what comes out of it...

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Gold Member
which means my post ##2## was just correct after all.

Mentor
which means my post ##2## was just correct after all.
Yes, except for the missing dv. My first quote in the previous thread came from your post #2.

chwala
Gold Member
ok just to finish on this...
on integration,i have
##ln x= -1.5 ln(3v+1)-0.5 ln (v-3)##

##lnx##= ln##\frac {(3v+1)^{-1.5}}{(v-3)^{-0.5}}##

##lnx##= ln##\frac {(v-3)^{0.5}}{(3v+1)^{1.5}}##

→##x##= ##\frac {(v-3)^{0.5}}{(3v+1)^{1.5}}##

→##x.\frac {(3x+y)^{1.5}}{(y)^{1.5}}##=##\frac {(x-3y)^{0.5}}{(y)^{0.5}}##

→##x^2.\frac {(3x+y)^3}{(y)^{1.5}}##=##\frac {x-3y}{y}##

looks a bit complicated...
on cross multiplication, we have two scenarios, either,
##y^3(x-3y)=0##
or
##x^2y(3x+y)^3=0##
tell me am doing something wrong...

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Gold Member
using the exact approach,

##M=3x+4y## and ##N=4x-3y## where ##Mdx+Ndy=0##

##\frac{∂M}{∂y}=4## and ##\frac {∂N}{∂x}=4## hence exact, therefore,

##U(x,y)= ∫(3x+4y)dx +∫(4x-3y)dy##

##c=\frac {3x^2}{2}+4xy-\frac {3y^2}{2}## where ##c## is an arbitrary constant.

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Mentor
ok just to finish on this...
on integration,i have
##ln x= -1.5 ln(3v+1)-0.5 ln (v-3)##
Plus the constant of integration...
chwala said:
##lnx##= ln##\frac {(3v+1)^{-1.5}}{(v-3)^{-0.5}}##

##lnx##= ln##\frac {(v-3)^{0.5}}{(3v+1)^{1.5}}##

→##x##= ##\frac {(v-3)^{0.5}}{(3v+1)^{1.5}}##

→##x.\frac {(3x+y)^{1.5}}{(y)^{1.5}}##=##\frac {(x-3y)^{0.5}}{(y)^{0.5}}##
I don't know about the step above. The substitution was y = vx, or v = y/x, so replace v with y/x. I can't tell if you did that.
chwala said:
→##x^2.\frac {(3x+y)^3}{(y)^{1.5}}##=##\frac {x-3y}{y}##

looks a bit complicated...
on cross multiplication, we have two scenarios, either,
##y^3(x-3y)=0##
or
##x^2y(3x+y)^3=0##
tell me am doing something wrong...

Mentor
##c=\frac {3x^2}{2}+4xy-\frac {3x^2}{2}## where ##c## is an arbitrary constant.
Exactly what I got.
Edit: That last term should be ##-\frac 3 2 y^2##.

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Gold Member
Plus the constant of integration...
I don't know about the step above. The substitution was y = vx, or v = y/x, so replace v with y/x. I can't tell if you did that.

aaaaargh stupid mistake on my part...i substituted ##x/y## instead of ##y/x## let me re do it...

Gold Member
##ln x= -1.5 ln(3v+1)-0.5 ln (v-3)##

##lnx##= ln##\frac {(3v+1)^{-1.5}}{(v-3)^{-0.5}}##

##lnx##= ln##\frac {(v-3)^{0.5}}{(3v+1)^{1.5}}##

→##x##= ##\frac {(v-3)^{0.5}}{(3v+1)^{1.5}}##

→##x.\frac {(3y+x)^{1.5}}{x^{1.5}}##=##\frac {(y-3x)^{0.5}}{(x)^{0.5}}##

→##x^2.\frac {(3y+x)^3}{x^3}##=##\frac {y-3x}{x}##

→##x^2.\frac {(3y+x)^3}{x^3}##=##\frac {y-3x}{x}##

→##\frac {(3y+x)^3}{x}##=##\frac {y-3x}{x}##

→##(3y+x)^3=y-3x##

Gold Member
Plus the constant of integration...
I don't know about the step above. The substitution was y = vx, or v = y/x, so replace v with y/x. I can't tell if you did that.
yeah i missed out on the constant as its an indefinite integral...noted...

Mentor
The first line in post #19 still omits the constant of integration. Also, the last line doesn't match your result in post #15. The solution you got in post #15, ##c=\frac {3x^2}{2}+4xy-\frac {3x^2}{2}##, satisfies the differential equation -- the one in post #19 does not.
I think you have a mistake in the work you did originally, but I can't put my finger on where you went wrong.

Gold Member
wait... but even in the exact approach, the first and third terms cancel out, leaving us with
##c=4xy##

Mentor
My bad. I misread your post #15. What I got was ##4xy + \frac 3 2 x^2 - \frac 3 2 y^2 = C## I have edited my earlier post.

Gold Member
i amended mine too, ...saw the error ...lol

Gold Member
now, i have re done the work for my post ##19## as follows,
##ln x= -1.5[ ln (3v+1)+0.5 ln (v-3)]##

##ln x + 1.5 ln (3v+1) +0.5 ln (v-3)=0##

##ln x + 1.5 ln (3v+1) +0.5 ln (v-3)= ln 1##

##\frac {(3y+x)^3} {x^3}####\frac {y-3x} {x}.x^2=0## on multiplying both sides by ##x^2##

##x(3y+x)(y-3x)=0##

##(3y^2-9xy+xy-3x^2)=0##

##(3y^2-8xy-3x^2)=0##

on dividing each term by ##2## and multiplying by ##-1##, we get,

##\frac {3x^2}{2}+4xy-\frac {3y^2}{2}##

does this make sense compared to the solution found by exact method?...i see an error in my working, i will look at this again...any insights will be appreciated.

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Gold Member
my other attempt on this...

##ln x= -1.5[ ln (3v+1)+0.5 ln (v-3)]+ln c##

##ln x + 1.5 ln (3v+1) +0.5 ln (v-3)=ln c##

##ln x + 1.5 ln (3v+1) +0.5 ln (v-3)= ln c##

##\frac{x(3y+x)^{1.5}(-3x+y)^{0.5}}{x^2}=c##

##\frac{(3y+x) (3y+x)^{0.5}(-3x+y)^{0.5}}{x}=c##
##\frac{(3y+x) (-9xy+3y^2-3x^2+xy)^{0.5}}{x}=c##
##\frac{(3y+x) (3y^2-8xy-3x^2)^{0.5}}{x}=c##

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Mentor
now, i have re done the work for my post ##19## as follows,
##ln x= -1.5[ ln (3v+1)+0.5 ln (v-3)]##
You can't omit the constant of integration! And you have a pair of brackets that shouldn't be there.
This should be ##\ln x= -1.5 \ln (3v+1)+0.5 \ln (v-3) + C##
chwala said:
##ln x + 1.5 ln (3v+1) +0.5 ln (v-3)=0##
Sign error in second term above.
Corrected the sign error and added in the constant.
##\ln x +1.5 \ln (3v+1) - 0.5 \ln (v-3) = C##
chwala said:
##ln x + 1.5 ln (3v+1) +0.5 ln (v-3)= ln 1##
Continue from my work above to
##\ln[\frac{x(3v + 1)^{3/2}}{(v - 3)^{1/2}} ] = C##
The next step will be
##\frac{x(3v + 1)^{3/2}}{(v - 3)^{1/2}} = e^C##
##e^C## is just another constant, call it A.
So you have ##\frac{x(3v + 1)^{3/2}}{(v - 3)^{1/2}} = A##
Replace v by y/x. Do not square both sides -- the factors in x will cancel, and things turn out as they should.
chwala said:
##\frac {(3y+x)^3} {x^3}####\frac {y-3x} {x}.x^2=0## on multiplying both sides by ##x^2##

##x(3y+x)(y-3x)=0##

##(3y^2-9xy+xy-3x^2)=0##

##(3y^2-8xy-3x^2)=0##

on dividing each term by ##2## and multiplying by ##-1##, we get,

##\frac {3x^2}{2}+4xy-\frac {3y^2}{2}##

does this make sense compared to the solution found by exact method?...i see an error in my working, i will look at this again...any insights will be appreciated.
No, there are a number of errors, as noted above. Also, you should end up with an equation; namely,
##\frac {3x^2}{2}+4xy-\frac {3y^2}{2} = C##

Gold Member
Mark after integration, aren't we supposed to have
##ln x= -1.5 ln (3v+1)-0.5 ln (v-3)+ln C##?..I spent time pondering over that.
giving me,
##ln x+1.5 ln (3v+1)+0.5 ln (v-3)=ln C##
and are we not supposed to consider the constant of integration as an ##ln## function i.e ## ln C##and not just a constant as you have indicated? or it does not matter?

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Mentor
Mark after integration, aren't we supposed to have
##ln x= -1.5 ln (3v+1)-0.5 ln (v-3)+ln C##?..I spent time pondering over that.
giving me,
##ln x+1.5 ln (3v+1)+0.5 ln (v-3)=ln C##
Yes, this looks fine, and is nearly the same as what you have in post #14.
However, the 2nd line in post #14 has an error.
##lnx##= ln##\frac {(3v+1)^{-1.5}}{(v-3)^{-0.5}}##
It should be ##\ln x = \ln[(3v + 1)^{-1.5}(v - 3)^{-0.5}] + \ln C##
##\Rightarrow \ln x = \ln \left( \frac{(3v + 1)^{-1.5}}{(v-3)^{0.5}}\right) + \ln C##
I have 3 pages of work, both sides, on this problem, and worked two different ways, so there are lots of opportunities to get a sign wrong, plus if you have a sign error in your work somewhere, and I continue from what you showed, that fouls things up as well.
chwala said:
and are we not supposed to consider the constant of integration as an ##ln## function i.e ## ln C##and not just a constant as you have indicated? or it does not matter?

chwala
Gold Member
let me check this again tomorrow, this problem has really mixed me up...

Mentor
From post #14:
on integration,i have ##ln x= -1.5 ln(3v+1)-0.5 ln (v-3)##
It took me a while to find it, but you have a mistake in the line above. Check your work for this integral, which produces the first term on the right side, above.
$$\int \frac{-3/2}{3v + 1}dv$$
To do this, use the substitution u = 3v + 1, so du = 3dv.
I think you'll find that you don't get ##-1.5 \ln(3v + 1)##.

chwala
Gold Member
so integrating that yields,
##-0.5 ln (3v+1)##
therefore, we now have
##ln x + 0.5 ln (3v+1)+0.5 ln (v-3)=ln C##
substituting ##y=vx## and simplifying leads to,
##(3y+x)(y-3x)=c##
##3y^2-9xy+xy-3x^2=c##
##3y^2-8xy-3x^2=c##
on multiplying both sides by ##-1## and dividing by ##2## yields,

##-\frac {3y^2}{2}##+##4xy##+ ##\frac {3x^2}{2}##

##\frac {3x^2}{2}##+##4xy## ##-\frac {3y^2}{2}## the desired result.

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Mentor
##3y^2-8xy-3x^2=c##
on multiplying both sides by ##-1## and dividing by ##2## yields,

##-\frac {3y^2}{2}##+##4xy##+ ##\frac {3x^2}{2}##

##\frac {3x^2}{2}##+##4xy## ##-\frac {3y^2}{2}## the desired result.
No, not quite. The "desired result" should be an equation,, as I wrote in post #27. ##\frac {3x^2}{2}##+##4xy## ##-\frac {3y^2}{2} = C##
The constant C that I wrote is equal to the constant that you wrote, c, multiplied by -1/2.

chwala
Gold Member
aaaaargh left out the constant again...nice day mark