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Thread: non-exact differential equation

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    non-exact differential equation

    at first trial , the dM/ dy not equal to dN / dx , so i multiply the whole equation with the integrating factor . so , i got M=4 , N= -cot y , dM/dy = 0 , dN/dx= 0 , so it's EXACT equation . But , my ans is wrong , the answer given is -(e^-4x) siny = C , which part of my working is wrong ?
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    Re: non-exact differential equation

    If it's $\displaystyle \begin{align*} 4\sin{(y)}\,\mathrm{d}x = \cos{(y)}\,\mathrm{d}y \end{align*}$ it's separable... Just write $\displaystyle \begin{align*} \int{4\,\mathrm{d}x } = \int{ \frac{\cos{(y)}}{\sin{(y)}}\,\mathrm{d}y} \end{align*}$ and integrate...
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    Re: non-exact differential equation

    Quote Originally Posted by Prove It View Post
    If it's $\displaystyle \begin{align*} 4\sin{(y)}\,\mathrm{d}x = \cos{(y)}\,\mathrm{d}y \end{align*}$ it's separable... Just write $\displaystyle \begin{align*} \int{4\,\mathrm{d}x } = \int{ \frac{\cos{(y)}}{\sin{(y)}}\,\mathrm{d}y} \end{align*}$ and integrate...
    ya , but i 'm asked to show it is a non-exact differential equation , and continue with exacft differential equation method
    Last edited by xl5899; Feb 15th 2016 at 04:27 AM.
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    Re: non-exact differential equation

    What answer are you expecting? I would have thought that it would be written as $4x - \ln \sin y = c$ or something similar. If is important to note that the solution of an exact equation is not an explicit function $u(x,y)$, it is an implicit function $u(x,y)=c$. The difference is very important. In the explicit function you have two free variables and the function returns a scalar. In the implicit function, there is only one free variable with the second determined by it's value.
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    Re: non-exact differential equation

    Quote Originally Posted by Archie View Post
    What answer are you expecting? I would have thought that it would be written as $4x - \ln \sin y = c$ or something similar. If is important to note that the solution of an exact equation is not an explicit function $u(x,y)$, it is an implicit function $u(x,y)=c$. The difference is very important. In the explicit function you have two free variables and the function returns a scalar. In the implicit function, there is only one free variable with the second determined by it's value.
    so , the correct solution would be showing the equation is EXACT first , then solve it using separable variable ?
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    Re: non-exact differential equation

    Quote Originally Posted by Archie View Post
    What answer are you expecting? I would have thought that it would be written as $4x - \ln \sin y = c$ or something similar. If is important to note that the solution of an exact equation is not an explicit function $u(x,y)$, it is an implicit function $u(x,y)=c$. The difference is very important. In the explicit function you have two free variables and the function returns a scalar. In the implicit function, there is only one free variable with the second determined by it's value.
    the answer given is -(e^-4x) siny = C
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    Re: non-exact differential equation

    No. Your answer is written as $u(x,y) = 4x + \ln \csc y$, but it should be $$\begin{aligned} u(x,y) = 4x + \ln \csc y &= c_1 &\text{and then} \\ \ln \mathrm e^{4x} - \ln \sin y &= c_1 \\ \ln {\mathrm e^{4x} \over \sin y} &= c_1 \\ {\mathrm e^{4x} \over \sin y} &= \mathrm e^{c_1} \\ \mathrm e^{-c_1} = -C &= \mathrm e^{-4x} \sin y \end{aligned}$$
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    Re: non-exact differential equation

    Any separable equation can easily (by "separating" it) put into "exact" form. If dy/dx= M(x)N(y). Separating dy/N(y)= M(x)dx which is the same as M(x)dx- (1/N(y))dy= 0. Since $\displaystyle \partial M(x)/\partial y= 0 = \partial N(y)/\partial x$.
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