Solve the initial value problem: y'=sqrt(xy), y(1)=4.

Begin by dividing both sides of the equation by sqrt(y).

This gives,

( dy /dx ) / sqrt(y) = sqrt(x),

and after multiplying by dx,

dy / sqrt(y) = sqrt(x) dx.

Now integrate both sides and get

S dy/sqrt(y) = S sqrt(x) dx ...

2*sqrt(y) = (2/3)*x*sqrt(x) + C.

The initial condition y(1)=4 gives a value for C, as follows,

2*sqrt(4) = (2/3)*1*sqrt(1) + C or 4=2/3+C, so C = 10/3.

Substituting gives the solution,

2 sqrt(y) = (2/3) x sqrt(x) + 10/3, or

3 sqrt(y) = x sqrt(x) + 5.