Gina
  • Gina
How about this one?,,,y'sinx=yIny
Mathematics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
This one is a bit more involved. It is again separable, so\[\frac{dy}{y.\ln(y)}=\frac{dx}{\sin(x)}\] The left-hand side can be integrated using the substitution u=ln(y), for then du=(1/y)dy and we can re-write the left-hand side as\[\frac{du}{u} \rightarrow \int\limits_{}{}\frac{du}{u}=\ln(u)+c_1\]Substituting u=ln(y) back in, we get\[\ln(\ln(y))+c_1\]as solution to the left-hand side. I'll post the right-hand side next...I have to break it up since this site is awkward to use when typing large pieces.
anonymous
  • anonymous
You have to use a t-substitution on the right-hand side. I was going to go into setting it up, but this site is difficult to use. I'm going to assume you have heard of the t-substitution (if not, it's easy to find out information). From this substitution technique, we have two results to use:\[\sin(x)=\frac{2t}{1+t^2}\] and \[dx=\frac{2dt}{1+t^2}\]Plugging these into dx/sin(x) we have\[\int_{}{}\frac{dx}{\sin(x)} \rightarrow \int\limits_{}{}\frac{1+t^2}{2t}.\frac{2dt}{1+t^2}=\int\limits_{}{}\frac{dt}{t}=\ln(t)+c_2\]Now, from the definition of t-substitution, \[t=\tan(\frac{x}{2})\]so upon substituting, the right-hand side becomes\[\ln(\tan(\frac{x}{2}))+c_2\]
Gina
  • Gina
nice i see)0

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Gina
  • Gina
am enjoying your leasons ,sir
anonymous
  • anonymous
Okay, now we bring the two answers together,\[\ln(\ln(y))+c_1=\ln(\tan(\frac{x}{2}))+c_2\]Collecting the constants on the RHS and exponentiating, we get,\[e^{\ln(\ln(y))}=e^{\ln(\tan(\frac{x}{2}))+(c_2-c_1)}\]Which simplifies to\[\ln(y)=e^{(c_2-c_1)}e^{\ln(\tan(\frac{x}{2}))}=A{\tan(\frac{x}{2})}\]where A is just e^(c_2-c_1), a constant. Exponentiating again isolates y,\[e^{\ln(y)}=e^{Atan(\frac{x}{2})}=e^Ae^{\tan(\frac{x}{2})}=Be^{\tan(\frac{x}{2})}\]where again, B is just the collected constant, e^A. So you're answer is \[y=Be^{\tan(\frac{x}{2})}\]
anonymous
  • anonymous
Grammatical error - *your* answer, not, *you're*.
anonymous
  • anonymous
Remember to check everything. You can do that by differentiating the final answer - you should be able to get back to the original equation. I haven't checked anything I've sent, so please do that. Good luck. Bed time...
anonymous
  • anonymous
PS - I'm glad you enjoyed the responses!
Gina
  • Gina
wow!!))thanx very much, sir,, i have really enjoyed your responses ,

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