Mathematics - Differential Equation Question with Solution | TestHub

$\left(x-x^3\right)dy=\left(y+y{x}^2-3x^4\right)dx$

$\Rightarrow xdy-ydx=\left(y{x}^2-3x^4\right)dx+x^{3}dy$

$\Rightarrow \frac{xdy-ydx}{x^2}=(ydx+xdy)-3x^{2}dx$

$\Rightarrow d\left(\frac{y}{x}\right)=d\left(xy\right)-d\left(x^3\right)$

Integrate

$\Rightarrow \frac{y}{x}=xy-x^3+c$

given $f\left(3\right)=3$

$\Rightarrow \frac{3}{3}=3\times 3-3^3+c$

$\Rightarrow c=19$

$\therefore \frac{y}{x}=xy-x^3+19$

at $x=4,\frac{y}{4}=4y-64+19$

$15y=4\times 45$

$\Rightarrow y=12$