Question
Verify these please
Verify that all members of the family y = (3) 1/2 (c - x2)-1/2 are solutions of the differential equation y' = (xy3)/3. Find a solution of the initial-value problem. Consider the following differential equation. Show that every member of the family of functions y = (3ln(x) + C)/x is a solution of the differential equation (Do this on paper. Your instructor may ask you to turn in this work.) Illustrate part (a) by graphing several members of the family of solutions on a common screen, (Do this on paper. Your instructor may ask you to turn in this work.) Find a solution of the differential equation that satisfies the initial condition y(1) = 8. (Enter the argument of the logarithmic function in parentheses.) Find a solution of the differential equation that satisfies the initial condition y(8) = 1. (Enter the argument of the logarithmic function in parentheses.)
Explanation / Answer
a) y = (3)^1/2 (c - x^2)^(-1/2)
=> y' = (-1/2)(-2x) y^3 / ((3^(1/2))^2 = (x*y^3)*3
b) y'/y^3 = x/3
=> -1/2y^2 = x^2 /6 + c
y(0) = 5
=> c = -1/50
=> (x^2)/3 + 1/y^2 + 1/25 = 0
