What have you tried so far?
Hey guys, i just started a new course in differential equations and im kinda having a hard time. Heres one of the problems I am having difficulty with:
Verify that the piecewise-defined function
is a solution of the differential equations on
Not quite. You cannot just choose two values of x. If x is any positive number, then so that y'= 2x. The equation xy'-2y = 0 becomes so the equation is satisfied for any positive x. If x is negative, . Do the same with that. Most importantly, what happens at x= 0? In particular what is the derivative of y at x= 0? (You can't just differentiate and set x= 0 because the derivative involves a limit from both sides: If x= 0 and h is positive, then x+h= 0+ h= h is positive, if h is negative, x+ h= h is negative.
If those two limits are the same, that is the dervative at x= 0. If they are not, the function is not differentiable at x= 0.