First slide

Methods of solving first order first degree differential equations

Question

$If a curve y=f(x) passes through the point (1,-1) and satisfies the differential equation y(1+xy)dx=xdy then$ $f (- \frac{1}{2}) is$

Moderate

A

$- \frac{4}{5}$
B

$\frac{2}{5}$
C

$\frac{4}{5}$
D

$- \frac{2}{5}$

Solution

$y d x + x y^{2} d x = x d y$
$y d x - x d y = - x y^{2} d x$
$\begin{array}{l} \int (\frac{ydx - xdy}{y^{2}}) = - \int xdx \\ \to \int d (x / y) = \frac{- x^{2}}{2} + c \\ \to \frac{x}{y} + \frac{x^{2}}{2} = c which passes through (1, - 1) \\ \to c = - \frac{1}{2} \\ \Rightarrow \frac{x}{y} + \frac{x^{2}}{2} = - \frac{1}{2} \\ \Rightarrow put x = - \frac{1}{2} \end{array}$
$\begin{array}{l} \Rightarrow \frac{- 1}{2 y} + \frac{1 / 4}{2} = - \frac{1}{2} \\ \Rightarrow \frac{- 1}{2 y} = - \frac{1}{8} - \frac{1}{2} = - \frac{5}{8} \\ \frac{- 1}{2 y} = - \frac{5}{8} \\ \Rightarrow y = \frac{4}{5} \end{array}$

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