If y = f(x) , and ( dy / dx ) + ( y / ( sqrt(a+(x^2)) ) ) = 0 I knew its solution is y = { sqrt(a+(x^2)) - x } , where a is a constant can any one give the proof , by solving the differntial equation. Are there any other solutions for the above given differential equation. I asked this other solutions because, on rearranging the given differential equation we get ( dy / y ) = - ( dx / ( sqrt(a+(x^2)) ) ) on integrating ln y = { integral ( - ( dx / ( sqrt(a+(x^2)) ) ) ) } + ln c so I may get the solution as y = ce^y1 , where y1 is funcion of x other than f(x) I am not sure about the existance of general solution , but I think it may exist. _____________________________________________________ so what I want is , 1. Solving procedure for the differntial equation to get the solution y = { sqrt(a+(x^2)) - x } 2. what is general answer for { integral ( - ( dx / ( sqrt(a+(x^2)) ) ) } 3. Is there any general solution for the differential equation given. other than y = { sqrt(a+(x^2)) - x } ______________________________________________________ Procedures I have tried and falied to do further. 1. we know d(sqrt(a+(x^2))) / dx = x / sqrt(a+(x^2)) so 1 / sqrt(a+(x^2)) = ( d(sqrt(a+(x^2))) / dx ) /x on substituting this value in the differential equation , we will get ( dy / dx ) + ( ( y * d( sqrt(a+(x^2)))/dx ) / x ) = 0 on solving this I got strucked at ln y = ( - ( sqrt(a+(x^2)) ) / x ) - { integral ( sqrt(a+(x^2)) / x^2) dx ) 2. rearranging the differntial equation we get ( dy / y ) = - ( dx / ( sqrt(a+(x^2)) ) ) take x = a cos(t) dx = - a sin(t) dt t = cos^-1 (x/a) on solving I got strucked at ( dy / y ) = { ( sqrt(2p) * sin(t) ) / sqrt( cos(2t) + 3 ) } dt I got no other ideas. I wish , I can get the answers for all the 3 questions I have asked.
Hi smslca, My calculus know-how stops at half-remembered integration techniques, so I can't help you directly with your query, but I can help with formulating your question. The [thread=42756]FAQ[/thread] includes a link to a [thread=61223]thread about using mimeTex[/thread] to post equations on Sciforums. To reformulate the first part of your post: Posted by smslca: If y = f(x) , and \(\frac {dy}{dx} + \frac {y}{\sqrt{a+x^2}} = 0\) I know its solution is \(y = \sqrt{a+x^2} - x\) , where a is a constant. can any one give the proof , by solving the differential equation. Are there any other solutions for the above given differential equation. I asked this other solutions because, on rearranging the given differential equation we get \(\frac{dy}{y} = - \frac{dx}{\sqrt{a+x^2}\) on integrating \(\ln {y} = -\int {\frac {dx}{\sqrt{a+x^2}}} + \ln {(c)}\) so I may get the solution as \(y = ce^{y_1}\) , where y1 is function of x other than f(x) I am not sure about the existence of a general solution , but I think it may exist. _____________________________________________________ so what I want is , 1. Solving procedure for the differential equation to get the solution \(y = \sqrt{a+x^2} - x \) 2. what is general answer for \( -\int\frac{dx}{\sqrt{a+x^2}\) 3. Is there any general solution for the differential equation given. other than \(y = \sqrt{a+x^2} - x \)Click the 'quote' button to see how I used the \( tags to make those equations.\)
Separation of variables is probably the easiest method, as mentioned in the OP. There's a simple, general method for solving linear first-order ODE's starting with the observation that \(\left[\mu(x)y\right]'=\mu(x)y'+\mu '(x)y\). You can also try the method of exact equations with integrating factors. All three of these techniques are considered standard in introductory ODE solving, so just about any intro textbook should cover them in the first few chapters.
As the first reply eludes to, the answer relates to hyperbolic functions, not trig functions. When you see a denominator of the form \(\sqrt{a^{2}+x^{2}}\) then you should think of the hyperbolic identity \(\cosh^{2}t - \sinh^{2}t = 1\), rather than the trig identity \(\cos^{2}t + \sin^{2}t = 1\). You can rearrange the former into the form you want, but not the latter.
Or you can go with the substitution \(x=a\tan\theta\) and use the identity \(1+\tan^2\theta=\sec^2\theta\). I can't personally remember seeing any integrals where it was absolutely necessary to resort to hyperbolic functions, but everyone has their own preferences.
Beauty is in the eye of the beholder. It is never necessary to do any sort of substitution (since you're just looking for an anti-derivative), but depending on which anti-derivatives are familiar to you, some substitutions are more helpful than others! There's a book I've forgotten the name of that does all the classic "hard" integrals that are ordinarily done using complex analysis, using only elementary real analysis methods. Some of the substitutions used in that text were outright bizarre! Please Register or Log in to view the hidden image!
