Answer:
y = 10(5x²+2)² -36
Step-by-step explanation:
(5x²+2)³[tex]\frac{dy}{dx}[/tex] - xy² = 0
(5x²+2)³[tex]\frac{dy}{dx}[/tex] = xy²
[tex]\frac{dy}{dx}[/tex] = [tex]\frac{xy^2}{(5x^2+2)^3}[/tex]
now we can do separation of variables
[tex]\frac{dy}{y^2} = \frac{xdx}{(5x^2+2)^3}[/tex]
[tex]\int\limits {\frac{dy}{y^2}} \, = \int\limits{\frac{xdx}{(5x^2+2)^3}}[/tex]
[tex]-\frac{1}{y} = -\frac{1}{10(5x^2+2)^2} +C[/tex] (I used u-sub to find the x integral, setting u = 5x^2+2)
y = 10(5x²+2)² + C
4 = 10(2)²+C
4 = 40 + C
C = -36
