Toronto Math Forum
MAT2442014F => MAT244 MathTests => Quiz 1 => Topic started by: Victor Ivrii on September 24, 2014, 10:17:48 PM

Please, post solution
2.6 p. 102, # 25
Solve
\begin{equation*}
(3x^2y+2xy+y^3)+ (x^2+y^2)y'=0.
\end{equation*}

\begin{equation}
(3 x^2 y + 2xy + y^3) + (x^2 + y^2) yâ€²= 0\label{A}
\end{equation}
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 $\mu = e^{3x}$
\begin{equation}
\bigl((3 x^2y + 2xy +y^3)e^{3x}\bigr)+ \bigl((x^2+y^2)e^{3x}\bigr)yâ€²= 0
\label{B}
\end{equation}
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,
\begin{equation}
\Psi (x, y) = (x^2+ \frac{1}{3} y^3)e^{3x} +C=0
\end{equation}$ is a solution.

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!

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!

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!

it is much easier to intergrate from N_{y} instead of M_{x}. But can someone teach me how to do that from M_{x}?