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Hello I was wondering why in this differential equation they did not use double angle identity, I understand how to get the other result also
So
1)Dif eqn:
Dy/dx +ycotx= cosx
Integrating factor IF= e^ (integrand of cotx) = sinx
Solution
yIF= integrand of QIF where Q is cotx from original question)
y sinx= integral of cosxsinx
Now I recognised this as ½ sin2x and the integral would be
ysinx= ½(-cos2x/2+ c)
therefore y= 1/sinx (-cos2x/4+ c/2)
however this could be done by substitution
y sinx= integral of cosxsinx let u=sinx to give
ysinx= integral of cosx./u du/cosx
ysinx= sin^2x +c
y=1/sinx (sin^2x +c)
why are they not the same ?? or are they really the same
I mean if we differentiate either sin^2x or -cos2x/2 we would get back to cosxsinx
2)
(x-2)dy/dx –y = (x-2)^3
dy/dx – y/(x-2)= (x-2)^2
IF= 1/(x-2)
yIF= intergral of QIF
y/(x-2)= integral of (x-2)^2 .1/(x-2)
y/(x-2)= integral of (x-2)
If I were to integrate (x-2) why is it (x-2)^2/2 +c is it the same as x^2/2-2x+c
(I am confused about this one as from expansion I can see its similar but if I put any value for c into either say 1 I get different answers)
3)Finally integral of 1/2x I can do this by dividing by a half
i.e ½ integral of 1/x =1/2 lnx
but is itthe same as 1/2ln2x as this wway I say what must I do to denominator (2x) to get to numerator and then simply multiply this coefficient to the natural log of the bottom
i.e 1/4x+3 = ½ ln 4x+3
Can I only do that if I cannot factorise out a coeff?
So
1)Dif eqn:
Dy/dx +ycotx= cosx
Integrating factor IF= e^ (integrand of cotx) = sinx
Solution
yIF= integrand of QIF where Q is cotx from original question)
y sinx= integral of cosxsinx
Now I recognised this as ½ sin2x and the integral would be
ysinx= ½(-cos2x/2+ c)
therefore y= 1/sinx (-cos2x/4+ c/2)
however this could be done by substitution
y sinx= integral of cosxsinx let u=sinx to give
ysinx= integral of cosx./u du/cosx
ysinx= sin^2x +c
y=1/sinx (sin^2x +c)
why are they not the same ?? or are they really the same
I mean if we differentiate either sin^2x or -cos2x/2 we would get back to cosxsinx
2)
(x-2)dy/dx –y = (x-2)^3
dy/dx – y/(x-2)= (x-2)^2
IF= 1/(x-2)
yIF= intergral of QIF
y/(x-2)= integral of (x-2)^2 .1/(x-2)
y/(x-2)= integral of (x-2)
If I were to integrate (x-2) why is it (x-2)^2/2 +c is it the same as x^2/2-2x+c
(I am confused about this one as from expansion I can see its similar but if I put any value for c into either say 1 I get different answers)
3)Finally integral of 1/2x I can do this by dividing by a half
i.e ½ integral of 1/x =1/2 lnx
but is itthe same as 1/2ln2x as this wway I say what must I do to denominator (2x) to get to numerator and then simply multiply this coefficient to the natural log of the bottom
i.e 1/4x+3 = ½ ln 4x+3
Can I only do that if I cannot factorise out a coeff?