Question

Let f be the function satisfying f'(x)=x√(f(x)) for all real numbers x, where f(3)=25. 1. Find f''(3).
2. Write an expression for y=f(x) by solving the differential equation dy/dx=x√y with the initial condition f(3)=25.

Please show work and reasoning.

Answer 1

y' = x√y
y'(3) = 3√25 = 15

y'' = √y + x/(2√y) y'
y''(3) = 5 + 3/10 * 15 = 9.5

y' = x√y
dy/√y = x dx
2√y = 1/2 x^2 + c
y(3) = 25, so
10 = 9/2 + c
c = 11/2

4√y = x^2+c
y = (x^2+11)^2/16
check: y' = x(x^2+11)/4 = x√y