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Solve the following differential equation : $ xdy-(y-x^3)dx=0 $

1 Answer

  • If the equation is of the form $\large\frac{dy}{dx}$$ + py= Q$, then the integrating factor is $e^{\large\int pdx}$
Step 1:
This can be written as
divide throughout by $x$
This is a linear differential equation of the form
Step 2:
The integrating factor I.F is
$e^{\int Pdx}=e^{-\large\frac{1}{x}dx}$
$\qquad=e^{\large -\log x}$
$\qquad=e^{\large \log (1/x)}$
Step 3:
The required solution is
$ye^{\int Pdx}=\int Qe^{\int Pdx}.dx+c$
$y.\big(\large\frac{1}{x})=$$-\int x^2\times \large\frac{1}{x}$$dx+c$
$\large\frac{y}{x}$$=-\int x\times dx+c$
(i.e) $x^3-2cx+2y=0$ is the required solution.
answered Nov 14, 2013 by sreemathi.v