Find the particular solution of the differential equation.

Question

$$ dy/dx = {x\sqrt{y}\cos(x)},(0,9) $$

This is what I have so far: $$ 2\sqrt{y} = x\sin(x)+\cos(x) $$ I don't know where to go from here.

Answer 1

$$dy/dx = {x\sqrt{y}\cos(x)},(0,9)$$ It's seperable $$\int \frac {dy}{\sqrt y} = \int {x\cos(x)}dx$$ $$\int \frac {dy}{\sqrt y} = x\sin(x)-\int \sin(x)dx$$ $$ 2\sqrt y = x\sin(x)+\cos(x)+K$$

Since $y(0)=9$

$$K= 2\sqrt 9 -1 =6 -1=5$$ $$ 2\sqrt y = x\sin(x)+\cos(x)+5$$

Answer 2

You will find

$$2\sqrt {y}=x\sin (x)+\cos (x)+C $$

and for $y=9$ and $x=0$, it gives

$$6=1+C $$ then ...