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Solve the differential equation$\Large \frac{dy}{dx}\normalsize =2xy-y$

$\begin{array}{1 1}(A)\;x=c\;e^{\large y-y^2} \\(B)\;y=c\;e^{\large x-x^2} \\ (C)\;c=xy \\(D)\;x\;e^{\large y-y^2}=c \end{array} $

1 Answer

  • If $\large\frac{dy}{dx}$$=f(x,y),$ where variables are seperable.
  • We express it in the form $f(x)dx=g(y)dy$
  • Then $\int f(x)dx=\int g(y)dy$
  • $\int \large\frac{dx}{x}$$=\log |x|+c$
Given $\large\frac{dy}{dx}$$=2xy-y$
This can be written as
On seperating the variables we get,
Integrating on both sides we get,
$\int\large\frac{dy}{y}$$=\int (2x-1)dx$
$\log _e y=\bigg( \large\frac{2x^2}{2}-x\bigg)+c$
=>$\log _e y=x^2-x+c$
Converting this to exporential form we get,
$c\;e^{\large x-x^2}=y$
Hence the required equation is
$y=c\;e^{\large x-x^2}$
answered May 8, 2013 by meena.p