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Solve the following $(x^{2}-yx^{2})dy+(y^{2}+xy^{2})dx=0$

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Toolbox:
  • First order , first degree DE
  • Variable separable : Variables of a DE are rearranged to separate then, ie
  • $f_1(x)g_2(y)dx+f_2(x)g_1(y)dy=0$
  • Can be written as $ \large\frac{g_1 (y)}{g_2(y)}$$dy=-\large\frac{f_1(x)}{f_2(x)}$$dx$
  • The solution is therefore $\int \large\frac{g_1(y)}{g_2(y)}$$dy=-\int \large\frac{f_1(x)}{f_2(x)}$$dx+c$
Step 1:
$(x^2-yx^2)dy+(y^2+xy^2)dx=0$
$x^2 (1-y) dy +y^2 (1+x) dx=0 $ is divided by $x^2y^2$
$\large\frac{1-y}{y^2}$$ dy+ \large\frac{1+x}{x^2}$$dx=0$
Step 2:
The variables are seperated
$\int (y^2 -\large\frac{1}{y})$$dy + \int (x^{-2} +\large\frac{1}{x})$$dx=c_1$
$-\large\frac{1}{y}$$-\log y-\large\frac{1}{x}$$+\log x =c_1$
$ \log \bigg( \large\frac{x}{y}\bigg)=\frac{1}{x}+\frac{1}{y}$$+c_1$
or $\large\frac{x}{y}$$=ce^{\Large\frac{1}{x}+\frac{1}{y}}$
$x= y ce^{\Large\frac{x+y}{x-y}}$

 

answered Sep 4, 2013 by meena.p
 
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