Author Topic: Q1 problem 1 (L5101)  (Read 3788 times)

Victor Ivrii

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Q1 problem 1 (L5101)
« on: September 24, 2014, 10:17:48 PM »

2.6 p. 102, # 25
Solve
\begin{equation*}
(3x^2y+2xy+y^3)+ (x^2+y^2)y'=0.
\end{equation*}
« Last Edit: September 24, 2014, 10:20:46 PM by Victor Ivrii »

Roro Sihui Yap

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Re: Q1 problem 1 (L5101)
« Reply #1 on: September 25, 2014, 12:31:06 AM »

(3 x^2 y + 2xy + y^3) + (x^2 + y^2) yâ€²= 0\label{A}

Let $M(x,y) = 3(x^2)y + 2xy + y^3$, $N(x,y) = x^2 + y^2$. Then $M_y(x,y) = 3x^2 + 2x + 3y^2$,  $N_x(x,y) = 2x$.

Equation (\ref{A}) is not exact. Lets try to find an integrating factor $\mu=\mu(x)$ to make it exact.
$d\mu/dx = (M_y - N_x)\mu / N\implies d\mu/dx = (3x^2 + 2x + 3y^2 - 2x)\mu / N \implies d\mu/dx = 3(x^2 + y^2) \mu / (x^2 + y^2)\implies d\mu/dx = 3Î¼\implies d\mu/ \mu = 3 dx\implies \ln \mu = 3x \implies \mu = e^{3x}$

Now multiply  the equation(\ref{A}) by  $\mu = e^{3x}$

\bigl((3 x^2y + 2xy +y^3)e^{3x}\bigr)+ \bigl((x^2+y^2)e^{3x}\bigr)yâ€²= 0
\label{B}

Now  $M(x,y) = 3(x^2y + 2xy +y^3)e^{3x}$,  $N(x,y) =(x^2+y^2)e^{3x}$. Then $M_y(x,y) = (3 x^2 + 2x + 3y^2)e^{3x}$, $N_x(x,y) = (2x3+x^2) + 3y^2)e^{3x}$,
$M_y(x,y) = N_x(x,y)$. Therefore the equation is exact. No need to check: it is exact due to construction of $\mu$. V.I.

There is a  $\Psi(x, y)$ such that:
\begin{gather}
\Psi _x(x, y) = M(x,y) =3(x^2y + 2xy +y^3)e^{3x},\label{C}\\
\Psi _y (x, y) = N(x,y) = (x^2+y^2)e^{3x}.\label{D}
\end{gather}

Integrating (\ref{C})  we have $\Psi (x, y) = (x^2+ \frac{1}{3} y^3)e^{3x} + f(y)$. Using this, differentiate to get
$\Psi_y (x, y) =(x^2+y^2)e^(3x) + f'(y)$. Easier to start from (\ref{D}) V.I.

Compare this with (\ref{D}): $f'(y) = 0$. $Meaning f(y) = C$, where $C$ is some constant
So,

\Psi (x, y) = (x^2+ \frac{1}{3}  y^3)e^{3x} +C=0
\$  is a solution.
« Last Edit: September 26, 2014, 09:40:20 AM by Victor Ivrii »

Chang Peng (Eddie) Liu

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Re: Q1 problem 1 (L5101)
« Reply #2 on: September 25, 2014, 01:04:58 AM »
I'm having trouble typing out equations with proper format in this forum, so I did it in MSW and screenshot the work.. Apologies in advanced!

Victor Ivrii

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Re: Q1 problem 1 (L5101)
« Reply #3 on: September 25, 2014, 02:12:34 AM »
Roro. I rewrote your solution using superior math typesetting by MathJax (Javascript LaTeX/MathML parser). Everyone can quote my solution to see how it is done. Also fix your name.

Eddie. You got a karma as an exception: no need to post inferior technically (screenshot) solution after superior (typed) and correct. You could export your solution to LaTeX and after minimal corrections post "typed" solution. However code would be ugly and difficult to edit.

For everyone in the future: red is mine!
« Last Edit: September 25, 2014, 02:17:03 AM by Victor Ivrii »

Chang Peng (Eddie) Liu

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Re: Q1 problem 1 (L5101)
« Reply #4 on: September 25, 2014, 12:40:57 PM »
Hi Prof. Ivrii,

It took me close to an hour to come up with that in MSW because I'm not used to typing out equations; so by the time I finished, Roro already posted it! But thank you!

Weiyang Guo

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Re: Q1 problem 1 (L5101)
« Reply #5 on: September 25, 2014, 10:51:20 PM »
it is much easier to intergrate from Ny instead of Mx. But can someone teach me how to do that from Mx?