Question

ydx+(1−3y)xy=3ydy

Answer 1

$ydx+(1−3y)xy=3ydy$

A first order linear differential equation in the form of $y'(x)+p(x)y=q(x)$

$y+(1-3y)xy=3y\frac{dy}{dx}$

Substitute $\frac{dy}{dx}=y'$

$y+(1-3y)xy=3yy'$

Rewrite,

$y'+xy=\frac{1}{3}+\frac{x}{3}$

Integration Factor : $IF = e^{\frac{x^2}{2}}$

Differential Equation: $(IF\times y)'= IF\times q(x)$

$(e^{\frac{x^2}{2}} \times y)'= e^{\frac{x^2}{2}} \times \frac{1}{3} +e^{\frac{x^2}{2}} \times \frac{x}{3}$

$y = \frac{\sqrt 2 \sqrt \pi erfi \frac{x}{\sqrt 2}}{6e^{\frac{x^2}{2}}}+\frac{1}{3}+\frac {c_1}{e^{\frac{x^2}{2}}}$