Find an explicit solution y(x), by using the given substitution then solving a linear ODE. dy dx SOLUTION: The chain rule gives du dx dy dx dy dx + y = y, y(0) X = X 1 du 3y dx X y=y=, y(1)=2. u= y. 3y2 dy. So dy dx dx du dx 11. y=y= du dx This can be solved by integrating factors. (x) = x-, so (x-u)' = 3x-4 + C So u = 1+ Cx-, and y = (1+ Cx-). Solving y(1) = 2 gives C = 9. 3u 3 u=y= >> + 2xy = xy, y(0)=1. u=y-. 1 du 3y dx du X - 3y = 3, dx 3 X So U = U=X 3 X x20 2 dy + xy + y = 0, dx y(1) = 2. u=y-.