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

Toolbox:
• If the equation is of the form $\large\frac{dy}{dx}$$+ py= Q, then the integrating factor is e^{\large\int pdx} Step 1: xdy-(y-x^3)dx=0 This can be written as xdy=(y-x^3)dx divide throughout by x \large\frac{dy}{dx}=\frac{y}{x}$$-x^2$
$\large\frac{dy}{dx}-\frac{y}{x}$$=-x^2 This is a linear differential equation of the form \large\frac{dy}{dx}$$+Py=Q$
Step 2:
The integrating factor I.F is
$e^{\int Pdx}=e^{-\large\frac{1}{x}dx}$
$\qquad=e^{\large -\log x}$
$\qquad=e^{\large \log (1/x)}$
$\qquad=\large\frac{1}{x}$
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}$$=-\int x\times dx+c \large\frac{y}{x}=\frac{-x^2}{2}$$+c$