1550078481-Ordinary_Differential_Equations__Roberts_

16 Ordinary Differential Equations

35. Verify that the piecewise defined function

{

0,

y(x) =

x2

'

x<O

0 :::: x

is differentiable on the interval ( -oo, oo) and is a solution of the differ-

ential equation (y')^2 - 4y = 0 on (-oo, oo ).

36. Verify that the piecewise defined function

{

x3, x<O

y( x ) =

0 , 0 :::: x

is differentiable on ( -oo, oo) and is a solution of the differential equation

(y')^2 - 9xy = 0 on (-oo, oo).

37. The differential equation

(L) y' = x3

is linear , but the differential equation

(N) (y')^2 = x6

is nonlinear.

a. Verify that the derivative of the piecewise defined function

{-x

4

/4 x<O

y(x) = '

x^4 /4, 0 :::: x

is

{-x'

x<O

y'(x) = '

x3 0 :::: x.

'

b. Show that y(x) is a solution on (-oo, oo) of (N) (y')^2 ::=:: x^6 , but

y( x) is not a solution on ( -oo, oo) of either (L) y' = x^3 or y' = -x^3.