Answer :
Answer:
[tex]y(x)=\sqrt[3]{sin(x)-3}[/tex]
Explanation:
The differential equation is:
[tex]\dfrac{dy}{dx}=\dfrac{cos(x)}{y^2}[/tex]
Separate variables:
[tex]y^2dy=cos(x)dx[/tex]
Integrate both sides:
[tex]\int y^2dy= \int xcos(x)dx[/tex]
[tex]\dfrac{y^3}{3}=sen(x)+C[/tex]
[tex]y^3=3sin(x)+C'[/tex]
Find C' using the inital condtion y(π/2) = 0
[tex]0=3sin(\pi /2)+C'\\\\0=3+C'\\ \\ C'=-3[/tex]
Then,
[tex]y^3=3sin(x)-3\\ \\ \\ y=\sqrt[3]{sin(x)-3}\\ \\ \\ y(x)=\sqrt[3]{sin(x)-3}[/tex]