Differential Equations

jsbeckton

Junior Member
Joined
Oct 24, 2005
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174
I'm studying my old tests for finals and have come across on that I got wrong and can't recall how to do it.

Show that the differential equation:

2xlnydx+(x2yy)dy=0\displaystyle 2x{\rm }\ln {\rm y dx + }\left( {\frac{{x^2 }}{y}{\rm - }\sqrt[{}]{{\rm y}}} \right){\rm dy = 0}

I've shown that My=Nx2xy=2xyequalityexact\displaystyle {\rm M}_{\rm y} {\rm = N}_{\rm x} {\rm } \Rightarrow {\rm }\frac{{{\rm 2x}}}{{\rm y}}{\rm = }\frac{{{\rm 2x}}}{{\rm y}}{\rm } \Rightarrow {\rm equality} \Rightarrow {\rm exact}

I'm having trouble remembering how to get the general solution:

x2lny23y32=c\displaystyle x^2 {\rm ln y - }\frac{{\rm 2}}{{\rm 3}}{\rm y}^{\frac{{\rm 3}}{{\rm 2}}} {\rm = c}

I've started with this:

\(\displaystyle \begin{array}{l}
g(x,y) = \int {2x{\rm }\ln y{\rm }dx} {\rm + c(y)} \\
{\rm = lny x}^{\rm 2} {\rm + c(y)} \\
\end{array}\)


Anyone know how to get this? Thanks.
 
Hello, jsbeckton!

I'm studying my old tests for finals and have come across on that I got wrong and can't recall how to do it.

Show that the differential equation:

\(\displaystyle 2x\cdot\ln y\cdot dx\,+\,\left( {\frac{x^2 }{y}\,-\,\sqrt{y}\right)dy\;=\;0\)

I've shown that: My=Nx        2xy=  2xy        equality      exact\displaystyle M_y\,=\,N_x\;\;\Rightarrow \;\;\frac{2x}{y}\,=\;\frac{2x}{y}\;\;\Rightarrow\;\;\text{equality }\;\Rightarrow\;\text{ exact}

I'm having trouble remembering how to get the general solution:   x2lny23y32=  C\displaystyle \;x^2\cdot\ln y\,-\,\frac{2}{3}y^{\frac{3}{2}}\:=\;C

I've started with this:

g(x,y)  =  Mdx  =  2xlnydx  =  x2lny+c1(y)\displaystyle g(x,y)\;=\;\int M\,dx\;=\;\int 2x\cdot \ln y\,dx\;=\;x^2\cdot\ln y\,+\,c_1(y) . . . . You're doing fine!
Now integrate N\displaystyle N with respect to y\displaystyle y.

\(\displaystyle \L g(x,y)\;=\;\int\left(\frac{x^2}{y}\,-\,y^{\frac{1}{2}}\right)\,dy\;=\;x^2\cdot\ln y\,-\,\frac{2}{3}y^{\frac{3}{2}}\,+\,c_2(x)\)


Now "combine" them: \(\displaystyle \L\;x^2\cdot\ln y\,-\,\frac{2}{3}y^{\frac{3}{2}} \;= \;C\)

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By the way, click on "Quote" and see the way I use LaTex.
. . It should have you hundreds of key-strokes.
 
See if this rings a bell.

M(x,y)=2xlnydx\displaystyle M(x,y)=2xlnydx

N(x,y)=(x2ysqrty)dy\displaystyle N(x,y)=(\frac{x^{2}}{y}-sqrt{y})dy

My=2xy=Nx\displaystyle \frac{\partial{M}}{\partial{y}}=\frac{2x}{y}=\frac{\partial{N}}{\partial{x}}

Exact, as you said.

fx=2xlny\displaystyle \frac{\partial{f}}{\partial{x}}=2xlny and fy=x2ysqrty\displaystyle \frac{\partial{f}}{\partial{y}}=\frac{x^{2}}{y}-sqrt{y}

(2xlny)dx=x2lny+g(y)\displaystyle \int_(2xlny)dx=x^{2}lny+g(y)

fy=d(x2lny+g(y))dy=x2y+g(y)\displaystyle \frac{\partial{f}}{\partial{y}}=\frac{d(x^{2}lny+g(y))}{dy}=\frac{x^{2}}{y}+g'(y)

gy=sqrty\displaystyle g'{y}=-sqrt{y} and g(y)=2y323\displaystyle g(y)=\frac{-2y^{\frac{3}{2}}}{3}

x2lny+g(y)=x2lny2y323=C\displaystyle x^{2}lny+g(y)=x^{2}lny-\frac{2y^{\frac{3}{2}}}{3}=C
 
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