Solve the initial value problem: y'=sqrt(xy), y(1)=4.
Begin by dividing both sides of the equation by sqrt(y).
( 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.