Question

the solution to the differential equation dy/dx=x^3/y^2, where y(2)=3 is?

Answer 1

\[\Large\rm \frac{dy}{dx}=\frac{x^3}{y^2},\qquad\qquad y(2)=3\]Separating variables gives us,\[\Large\rm y^2~dy=x^3~dx\]Integrating,\[\Large\rm \int\limits y^2~dy=\int\limits x^3~dx\]gives us,\[\Large\rm \frac{1}{3}y^3=\frac{1}{4}x^4+c\]

Answer 2

From here, use your initial data to solve for c.
\(\Large\rm y=3\quad when\quad x=2\)

\[\Large\rm \frac{1}{3}(3)^3=\frac{1}{4}(2)^4+c\]

Answer 3

After you've gotten your c value, plug it in,
and then solve your equation explicitly for y.\[\Large\rm y=\sqrt[3]{\frac{3}{4}x^4+3c}\]

Answer 4

thank you