Solve the given initial value problem.

Question

5t(dy/dx)=(y^2)-y, y(1)=-7

Let y(t) be the solution of the initial value problem. Find the value of y(7^5) with the answer rounded to 4 digits after the decimal point.

Answer 1

5tdy/dt = y² - y; 5(1/(y-1) - 1/y)dy = dt/t; ∫5(1/(y - 1) - 1/y)dy = ∫dt/t; 5lnIy-1/yI = lnItI + 5lnC;

(y - 1)/y = Ct^1/5; At t = 1 we have - 8/(- 7) = C; C = 8/7

1 - 1/y = 8/7t^1/5; 1 - 1/y(7⁵) = 8/7·7; 1/y(7⁵) = - 7; y(7⁵) = - 1/7 = - 0.1429