Answer to Question #175146 in Differential Equations for rabi

Question

dy/dx =x(ex² +2)/6y²

Answer 1

${dy \over dx} ={x(e^{x^2} +2) \over 6y^2}$

$\int6y^2dy=\int xe^{x^2} dx+ \int 2x dx$

To integrate $\int xe^{x^2} dx$

Let ${x^2} =u, then$

${du \over dx}=2x$

${du \over 2} = xdx$

${1 \over 2} \int e^ u du= {e^{x^2} \over 2} +c$

$\therefore \int6y^2dy=\int xe^{x^2} dx+ \int 2x dx$

$2y^3={e^{x^2} \over 2} + x^2+c$

$y=$ $({e^{x^2} \over 4} + {x^2 \over 2} + {c \over 2})^{1 \over 3}$