2 a Show that x x2 is an explicit solution to x dy dx 2y

2) (a) Show that (x) = x² is an explicit solution to x (dy / dx) = 2y on the interval (-,).

(b) Show that (x) = e^x - x is an explicit solution to dy / dx + y²= e^2x + (1 - 2x)e^x + x² - 1 on the interval (-,).

(c) Show that (x) = x² - x^-1 is an explicit solution to x²d²y / dx² = 2y on the interval (0, ).

Solution

a) x (dy / dx) = 2y

=> dy/y = 2 dx/x

Integrate both side,

ln y = 2*ln x.c, where c is constant.

=> ln y = ln x² + ln d, where d is another constant.

For explicit solution, make contant term 0.

=> ln y = ln x²

Take anti-log both sides, which gives

y = x² = (x)

Hence (x) = x² is an explicit solution to x (dy / dx) = 2y.

****not mentioned to do all questions. so doing just first one.